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+\begin{thebibliography}{10}
+
+\bibitem{BaezaTrees}
+R.~A. Baeza-Yates.
+\newblock On embedding a binary tree on a hypercube.
+\newblock Submitted for publication, November 1988.
+
+\bibitem{Tables}
+R.~J. Beach.
+\newblock {\em Setting Tables and Illustrations with Style}.
+\newblock PhD thesis, University of Waterloo, 1985.
+
+\bibitem{Karlsruhe}
+A.~Br{\"u}ggemann-Klein, P.~Dolland, and A.~Heinz.
+\newblock How to please authors and publishers: A versatile document
+  preparation system at {K}arlsruhe.
+\newblock In J.~D\'esarm\'enien, editor, {\em {\TeX{}} for Scientific
+  Documentation}, pages 8--31, Strasbourg, France, June 1986.
+\newblock Springer-Verlag Lecture Notes in Computer Science~236.
+
+\bibitem{Exeter}
+A.~Br{\"u}ggemann-Klein and D.~Wood.
+\newblock Drawing trees nicely with {\TeX}.
+\newblock In M.~Clark, editor, {\em Proceedings of the Third European
+  \TeX{}-Conference}, Exeter, England, July 1988.
+\newblock To appear.
+
+\bibitem{BinarySearchTrees}
+L.~Devroye.
+\newblock A note on the height of binary search trees.
+\newblock {\em Journal of the ACM}, 33(3):489--498, July 1986.
+
+\bibitem{BinaryTrees}
+Ph. Flajolet and A.~Odlyzko.
+\newblock The average height of binary trees and other simple trees.
+\newblock {\em Journal of Computer and System Sciences}, 25:171--213, 1982.
+
+\bibitem{DocumentFormatting}
+R.~Furuta, J.~Scofield, and A.~Shaw.
+\newblock Document formatting systems: Surveys, concepts, issues.
+\newblock {\em Computing Surveys}, 14(3):417--472, 1982.
+
+\bibitem{GML}
+Ch.~F. Goldfarb.
+\newblock A generalized approach to document markup.
+\newblock {\em SIGPLAN Notices of the ACM}, 16(6):68--73, June 1981.
+
+\bibitem{KWIFIP}
+R.~Klein and D.~Wood.
+\newblock On binary trees.
+\newblock In G.~Ritter, editor, {\em Proceedings of the 11th World Computer
+  Congress}, San Francisco, USA, 1989.
+\newblock To appear.
+
+\bibitem{ACP}
+D.~E. Knuth.
+\newblock {\em Fundamental Algorithms}, volume~1 of {\em The Art of Computer
+  Programming}.
+\newblock Addison-Wesley, Reading, Massachusetts, 1973.
+
+\bibitem{TeXbook}
+D.~E. Knuth.
+\newblock {\em The {\TeX{}}book}, volume~A of {\em Computers \& Typesetting}.
+\newblock Addison-Wesley, Reading, Massachusetts, 1986.
+
+\bibitem{LaTeX}
+L.~Lamport.
+\newblock {\em {\LaTeX}, User's Guide \& Reference Manual}.
+\newblock Addison-Wesley, Reading, Massachusetts, 1986.
+
+\bibitem{OAPD}
+Th. Ottmann and P.~Widmayer.
+\newblock {\em Algorithmen und Datenstrukturen}.
+\newblock Bibliographisches Institut, Mannheim, 1989.
+\newblock To appear.
+
+\bibitem{Grif}
+V.~Quint, I.~Vatton, and H.~Bedor.
+\newblock Grif: An interactive environment for {\TeX{}}.
+\newblock In J.~D\'esarm\'enien, editor, {\em {\TeX{}} for Scientific
+  Documentation}, pages 145--158, Strasbourg, France, June 1986.
+\newblock Springer-Verlag Lecture Notes in Computer Science~236.
+
+\bibitem{Scribe}
+B.~K. Reid.
+\newblock {\em Scribe: A Document Specification Language and its Compiler}.
+\newblock PhD thesis, Carnegie Mellon University, 1980.
+
+\bibitem{TidierTrees}
+E.~M. Reingold and J.~S. Tilford.
+\newblock Tidier drawings of tree.
+\newblock {\em IEEE Transactions on Software Engineering}, 7(2):223--228, March
+  1981.
+
+\bibitem{Complexity}
+K.~J. Supowit and E.~M. Reingold.
+\newblock The complexity of drawing trees nicely.
+\newblock {\em Acta Informatica}, 18(4):377--392, 1983.
+
+\bibitem{TidyTrees}
+Ch. Wetherell and A.~Shannon.
+\newblock Tidy drawings of trees.
+\newblock {\em IEEE Transactions on Software Engineering}, 5(5):514--520,
+  September 1979.
+
+\end{thebibliography}
+
diff --git a/Master/texmf-dist/doc/plain/treetex/epodd.tex b/Master/texmf-dist/doc/plain/treetex/epodd.tex
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+% This is epodd.tex, the description of the treetex macro package as it will
+% appear in EP-ODD in summer 89. It is in some aspects more general
+% than tree_doc.tex and corrects an error in the computation of
+% the number of registers used by treetex. The user interface of
+% treetex is explained in more detail in tree_doc.tex.
+
+\documentstyle[12pt,fullpage]{article}
+
+\clubpenalty=10000
+\widowpenalty=10000
+
+\def\addcontentsline#1#2#3{\relax}% Some captions are too long for some
+     % TeX installations (buffer size too small)
+
+\newenvironment{lemma}{\begingroup\samepage\begin{lemmma}\ }{\end{lemmma}%
+     \endgroup}
+\newtheorem{lemmma}{Lemma}[section]
+\newenvironment{proof}{\begin{prooof}\rm\ \nopagebreak}{\end{prooof}}
+\newcommand{\proofend}{\qquad\ifmmode\Box\else$\Box$\fi}
+\newtheorem{prooof}{Proof}
+\renewcommand{\theprooof}{}   % makes shure that prooof doesn't get numbers
+\newenvironment{Figure}{\begin{figure}\vspace{1\baselineskip}}%
+                       {\vspace{1\baselineskip}\end{figure}}
+\newlength{\figspace}         % space between figures in a single
+\setlength{\figspace}{30pt}   % Figure environment
+
+\newcommand{\var}[1]{{\it #1\/}}     % use it for names of variables
+\renewcommand{\emph}[1]{{\em #1\/}}    % use it for emphazided text
+                                     % (This notion sticks to the
+                                     % applicative style of markup.)
+\renewcommand{\O}{{\rm O}}           % O-notation, also for math mode
+\newcommand{\T}{{\cal T}}            % the set T in math mode
+\newcommand{\TreeTeX}{Tree\TeX}
+\newcommand{\fig}[1]{Figure~\ref{#1}}
+\let\p\par
+
+\input treetex
+\Treestyle{\vdist{20pt}\minsep{16pt}}
+\dummyhalfcenterdim@n=2pt
+
+% \def\Tree#1\end#2{\end{Tree}}  % Trees are not processed
+% \let\endTree\relax                  %
+
+\def\Node(#1,#2){\put(#1,#2){\circle*{4}}}
+\def\Edge(#1,#2,#3,#4,#5){\put(#1,#2){\line(#3,#4){#5}}}
+
+\def\enode{\node{\external\type{dot}}}
+\def\inode{\node{\type{dot}}}
+
+\def\e{\node{\external\type{dot}}}
+\def\i{\node{\type{dot}}}
+\def\il{\node{\type{dot}\leftonly}}
+\def\ir{\node{\type{dot}\rightonly}}
+
+\newcommand{\stack}[3]{%
+     \vtop{\settowidth{\hsize}{#1}%
+     \setlength{\leftskip}{0pt plus 1fill}%
+     \setlength{\baselineskip}{#2}#3}}
+
+\let\multic\multicolumn
+
+\newlength{\hd} % hidden digit
+\setbox0\hbox{1}
+\settowidth{\hd}{\usebox{0}}
+\newcommand{\ds}{\hspace{\hd}} % digit space
+
+\newcommand{\ccol}[1]{\multicolumn{1}{c}{#1}}
+
+\hyphenation{post-or-der sym-bol Karls-ruhe bool-ean}
+
+\begin{document}
+
+\bibliographystyle{plain}
+
+\title{Drawing Trees Nicely with \TeX\thanks{This work was supported by
+     a Natural Sciences and Engineering Research Council of Canada
+     Grant~A-5692, a Deutsche Forschungsgemeinschaft Grant~Sto167/1-1,
+     and a grant from the Information Technology Research  Centre.
+     It was begun during the first author's stay with
+     the Data Structuring Group in Waterloo.}}
+\author{Anne Br\"uggemann-Klein\thanks{Institut f\"ur Informatik,
+     Universit\"at Freiburg, Rheinstr.~10--12, 7800~Freiburg,
+     West~Germany}\ \and Derick Wood\thanks{Data
+     Structuring Group, Department of Computer Science, University of
+     Waterloo, Waterloo, Ontario N2L~3G1, Canada}}
+\date{}
+\maketitle
+
+\begin{abstract}
+
+We present a new solution to the tree drawing problem that
+integrates an excellent tree drawing algorithm into one of the best text
+processing systems available. More precisely, we present a \TeX{} macro package
+called \TreeTeX{} that produces drawings of trees from a purely logical
+description. Our approach has three advantages: Labels
+for nodes can be handled in a reasonable way; porting
+\TreeTeX{} to any site running \TeX{} is a trivial operation; and
+modularity in the description of a tree and \TeX{}'s macro capabilities
+allow for libraries of subtrees and tree classes.
+
+In addition, \TreeTeX{} has an option that produces
+drawings that make the
+\emph{structure} of the trees more obvious to the human eye,
+even though they may not be as aesthetically pleasing.
+
+\end{abstract}
+
+\section{Introduction}
+
+The problem of successfully integrating pictures and text in a
+document processing environment is tantalizing and difficult.
+Although there are systems available that allow such integration, they
+fall short in many ways, usually in document quality. Furthermore,
+most authors using document  preparation systems are neither book 
+designers nor graphic artists.  Just as modern document preparation
+systems do not expect an author to be a book designer, so we would
+prefer that they do not expect an author to be a graphic artist. The
+second author, Wood, needed to draw many trees in a series of papers
+on trees and in a projected book on trees. This problem enabled us
+to tackle the integration issue for one subarea of graphics, namely,
+tree drawing. We had the decided advantage that there already existed good
+algorithms to draw trees {\em without any author intervention}.
+Previous experience of the integration of pictures and text had been
+uninspiring; the  systems expected the author to prepare each picture
+in total. For example, a tree could be built up from smaller
+subtrees but the relative placement of them was left to the author.
+This situation continues to hold today with the drawing facilities
+available on most personal computers, and, because of this, the
+resulting figures still appear to be ``hand-drawn.'' Additionally,
+they are of inferior quality when compared with the quality of
+the surrounding text.
+
+In this paper we present an entirely new solution that integrates
+a tree drawing algorithm into one of the best text processing
+systems available. More precisely, we  describe \TreeTeX{}, a
+\TeX{} macro package that produces an aesthetically pleasing
+drawing of a tree from a purely logical description.
+We made two fundamental design
+decisions that heavily influenced the method of implementation.
+First, we wanted to allow an author to label the nodes of a tree.
+This decision means that the tree drawing package must be able to
+typeset labels exactly as they would be typeset by the typesetting
+program. There are two reasons for this. Text should be typeset
+consistently, wherever it appears in a document, and the tree
+drawing program needs to know the dimensions of the typeset labels.
+Second, we wanted to ensure that the program could be ported
+easily to other installations and sites, so that other, putative
+users would be able to use it easily.
+Indeed, \TreeTeX{} has been used successfully to typeset trees in
+\cite{BaezaTrees}, \cite{KWIFIP}, and \cite{OAPD}.
+
+By basing our package on \TeX{}, which for more subjective
+reasons we preferred over other typesetting systems such as
+troff, we could ensure wide interest
+in the package. By implementing it as a \TeX{} macro package
+instead of a preprocessor
+we made porting trivial and, furthermore, this also ensured
+consistency of typeset text within a document.
+The down side of this decision is that we had  to program with
+\TeX{} macros, not an experience to be recommended, and we had to live
+with the inherent register limitations of \TeX{}.
+
+This paper consists of a further nine  sections. In Sections~2, 3 and~4,
+we discuss the aesthetics of tree drawing and the algorithm of
+Reingold and Tilford~\cite{TidierTrees}. In Sections~5, 6, and~7, we
+describe our method of incorporating tree drawing into \TeX{}. Then,
+in the last three short sections, we consider the expected number of
+registers \TeX{} needs to draw a tree, the user interface (and three
+\TreeTeX{} examples), and discussion of, among other things, the
+performance  of \TreeTeX{}.
+
+\section{Aesthetical criteria for drawing trees}
+
+In this paper, we are dealing with ordered
+trees in the sense of~\cite{ACP}, specifically binary and unary-binary
+trees. A {\em binary tree\/} is a finite set of nodes that either
+is empty, or consists of a root and two disjoint binary trees called
+the left and right subtrees of the root. A {\em unary-binary tree\/} is
+a finite set of nodes that either is empty, or consists of a root and
+two disjoint unary-binary trees, or consists of a root and one
+nonempty unary-binary tree. An {\em extended binary tree\/} is a binary tree
+in which each node has either two nonempty subtrees or two
+empty subtrees.
+
+There are some basic agreements on how such trees should be drawn, reflecting
+the top-down and left-right ordering of nodes in a tree.
+In \cite{TidierTrees} and \cite{TidyTrees} these basic agreements were
+formalized as the following axioms.
+
+\begin{enumerate}
+\item[1.] Trees impose a distance on the nodes; no node
+          should be closer to the root than any of its
+          ancestors.
+\item[2.] Nodes on the same level should lie on a straight
+          line, and the straight lines defining the levels should be
+          parallel.
+\item[3.] The relative order of nodes on any level should be the same
+          as in the level order traversal of the tree.
+\end{enumerate}
+
+These axioms guarantee that trees are drawn as planar graphs: edges do
+not intersect except at nodes. Two further axioms improve the aesthetical
+appearance of trees.
+
+\begin{enumerate}
+\item[4.] In a unary-binary tree, each left child should be positioned
+          to the left of its parent, each
+          right child to the right of its parent, and each unary child
+          should be positioned below its parent.
+\item[5.] A parent should be centered over its children.
+\end{enumerate}
+
+An additional axiom deals with the problem of tree drawings becoming too wide
+and therefore exceeding the physical limit of the output medium:
+
+\begin{enumerate}
+\item[6.] Tree drawings should occupy as little width as possible without
+          violating the other axioms.
+\end{enumerate}
+
+In \cite{TidyTrees}, Wetherell and Shannon introduce two algorithms for
+tree drawings, the first of which fulfills axioms~1--5, and the second
+1--6. However, as Reingold and Tilford in \cite{TidierTrees}
+point out, there is a lack of symmetry in the algorithms of
+Wetherell and Shannon which may lead to unpleasant results;
+therefore, Reingold and Tilford introduce a new structured
+axiom.
+
+\begin{enumerate}
+\item[7.] A subtree of a given tree should be
+          drawn the same way regardless of where it occurs in the tree.
+\end{enumerate}
+
+Axiom~7 allows the same tree to be drawn differently only when it occurs as
+a subtree in different trees.
+Reingold and Tilford give an algorithm which fulfills axioms~1--5
+and~7. Although
+this algorithm doesn't fulfill axiom~6,
+the aesthetical improvements are well worth the additional space.
+\fig{algorithms} illustrates the benefits of axiom~7, and \fig{narrowtrees}
+shows that the algorithm of Reingold and Tilford violates axiom~6.
+
+\begin{Figure}
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\node{\external\type{dot}\rght{\unskip\hskip2\mins@p\hskip2\dotw@dth}}
+\enode\enode\inode\enode\enode\inode\inode\inode
+\inode
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\inode
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\caption{The left tree is drawn by the algorithm of Wetherell and Shannon,
+and the tidier right one is drawn by the algorithm of Reingold and Tilford.}
+\label{algorithms}
+
+\vspace{\figspace}
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\enode\enode\enode\enode\enode\enode\enode\enode\enode
+\enode\inode\inode\inode
+\enode\inode\inode\inode
+\enode\inode\inode\inode
+\enode\inode\inode\inode
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\enode\enode\enode\enode\enode\enode\enode\enode
+\node{\external\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\inode
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\caption{The left tree is drawn by the algorithm of Reingold and Tilford, but
+the right tree shows that narrower drawings fulfilling all aesthetic axioms
+are possible.}
+\label{narrowtrees}
+\end{Figure}
+
+
+\section{The algorithm of Reingold and Tilford}
+
+The algorithm of Reingold and Tilford (hereafter called ``the RT~algorithm'')
+takes a modular approach to the
+positioning of nodes. The relative positions of the nodes in a subtree
+are calculated independently of the rest of the tree. After the
+relative positions of two subtrees have been calculated, they can be
+joined as siblings in a larger tree by placing them as close
+together as possible and centering the parent node above them.
+Incidentally, this modular approach is the reason that the
+algorithm fails to fulfill axiom~6; see~\cite{Complexity}.
+Two sibling subtrees are placed as close together as possible,
+during a postorder traversal, as follows.
+Imagine that the two subtrees of a binary node
+have been drawn and cut out of paper along
+their contours. Then, starting with the two subtrees superimposed at their
+roots, move them apart until a minimal agreed upon distance
+between the trees is obtained at each level. This can be done gradually.
+Initially, their roots are separated by some agreed upon minimum
+distance; then, at the next level, they are pushed
+apart until the minimum separation is established there.
+This process is continued at successively lower levels until the
+last level of the shorter subtree is reached. At some levels no movement may be
+necessary, but at no level are the two subtrees moved closer
+together. When the process is complete, the position of the
+subtrees is fixed relative to their parent, which is centered over them.
+Assured that the subtrees will never be placed closer together,
+the postorder traversal is continued.
+
+A nontrivial implementation of
+this algorithm has been obtained by Reingold and Tilford in~\cite{TidierTrees} 
+that runs in time $\O(N)$, where $N$ is the number of
+nodes of the tree to be drawn.
+Their crucial idea is to keep track of the contour of the subtrees
+by special pointers, called threads, such that whenever
+two subtrees are joined, only the
+top part of the trees down to the lowest level of the
+smaller tree need to be taken into account.
+
+The nodes are positioned on a fixed grid and are
+considered to have zero width; labeling is not provided. 
+Although the algorithm only draws binary trees, it is easily 
+extended to multiway trees.
+
+\section{Improving human perception of trees}
+
+It is common understanding in book design that aesthetics and readability
+don't necessarily coincide, and---as Lamport (\cite{LaTeX}) puts it---%
+``documents are meant to be read, not hung in museums.'' 
+Therefore, readability is more important than aesthetics.
+
+When it comes to tree drawings, readability means that the structure of
+a tree must be easily recognizable. This criterion is not always met
+by the RT~algorithm. As an example, there are trees whose structure is 
+different even though they have the same number
+of nodes on each level. The RT~algorithm might assign identical positions to
+these nodes making it very hard to perceive the structural differences.
+Hence, we have modified the RT~algorithm such that additional white space
+is inserted between subtrees of
+\emph{significant} nodes. Here a binary node
+is called significant if the minimum distance
+between its two subtrees is achieved \emph{below} their root level.
+Setting the amount of additional white space to zero retains the original RT~%
+placement. The effect of having nonzero additional white space between
+the subtrees of significant
+nodes is illustrated in \fig{addspace}.
+
+Another feature we have added to the RT~algorithms is the possibility to draw
+an unextended binary tree with the same placement of nodes as its
+associated extended version;
+this makes the structure of a tree more prominent; see \fig{extended}.
+We define the \emph{associated extended version}
+of a binary tree to be the binary tree obtained by replacing each empty subtree
+having a nonempty sibling with a subtree consisting of one node. 
+
+\begin{Figure}
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\e\il\e\e\i\i\il % the left subtree
+\e\ir\il % the right subtree
+\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\il\il\il % the left subtree
+\e\e\i\e\i\il % the right subtree
+\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\adds@p10pt
+\begin{Tree}
+\e\il\e\e\i\node{\type{dot}\lft{$\longrightarrow$}}\il % the left subtree
+\e\ir\il % the right subtree
+\node{\type{dot}\lft{$\longrightarrow$}}
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\il\il\il % the left subtree
+\e\e\i\e\i\il % the right subtree
+\node{\type{dot}\lft{$\longrightarrow$}}
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\adds@p0pt
+
+\caption{The nodes of the first two trees are placed in the same positions
+by the RT~algorithm, although the structure of the two trees is different.
+The alternative drawings highlight the structural differences 
+of the trees by adding additional white space between the subtrees of
+($\longrightarrow$) significant nodes.}
+\label{addspace}
+\end{Figure}
+
+\begin{Figure}
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\e\e\i\il\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\hbox{}\qquad
+\begin{Tree}
+\e\e\i\e\i\e\ir\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\hbox{}\\
+\extended
+\begin{Tree}
+\e\e\i\il\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\hbox{}\qquad
+\begin{Tree}
+\e\e\i\e\i\e\ir\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\hbox{}\\
+\noextended
+\begin{Tree}
+\e\e\i\e\i\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\caption{As in the previous figure, the nodes of the first two trees
+are placed in the same position by the RT algorithm,
+although their structure is different. The modified
+RT~algorithms highlights the structural differences of the trees by 
+drawing them like their identical extended
+version (given in the third row), but suppressing the additional nodes.}
+\label{extended}
+\end{Figure}
+
+
+\section{Trees in a document preparation environment}
+
+Drawings of trees do not usually appear by themselves,
+but are included in some text
+that is itself typeset by a text processing system. Therefore, a typical
+scenario is a pipe of three stages. First, we have a tree drawing
+program that calculates the positioning of the nodes of the tree to
+be drawn and outputs a description of the tree drawing in
+some graphics language; this is followed by
+a graphics system that transforms this
+description into an intermediate language that can be interpreted by the output
+device; and, finally, we have the
+text processing system that integrates the output of the
+graphics system into the text.
+
+This scenario loses its linear structure once nodes have to be labeled, since
+the labeling influences the positioning of the nodes. Labels usually occur
+inside, to the left of, to the right of, or beneath nodes (the latter only for
+external nodes). Their widths should certainly be taken into account
+by the  tree drawing algorithm. But the labels have to be typeset first
+to determine their extensions,
+preferably by the typesetting program that
+is used for the regular text, because this ensures uniformity in the textual
+parts of the document and provides the author with the full power of a
+text processing system for composing the labels. Hence, a more complex
+communication scheme than a simple pipe is required.
+
+Although a system of two processes running simultaneously might be the most
+elegant solution, we wanted a system that is easily portable to
+widely different machines at our sites
+including personal computers with single process
+operating systems.
+Therefore, we decided to use a text processing system
+having programming facilities powerful enough to 
+program a tree drawing algorithm and graphics facilities powerful enough
+to draw a tree. One text processing system
+rendering outstanding typographic quality and satisfactory programming
+facilities is \TeX, developed by Knuth at Stanford University;
+see~\cite{TeXbook}.
+The \TeX{} system includes the following programming facilities.
+
+\begin{enumerate}
+\item[1.] Datatypes:\\
+     integers~(256), dimensions\footnote{The term \emph{dimension} is used
+     in \TeX\ to describe physical measurements of typographical objects;
+     for example, the length of a word.}~(512), 
+     boxes~(256), tokenlists~(256), and
+     boolean variables~(unrestricted).
+\item[2.] Elementary statements:\\
+     $a:=\rm const$, $a:=b$ (all types);\\
+     $a:=a+b$, $a:=a*b$, $a:=a/b$ (integers and dimensions); and\\
+     horizontal and vertical nesting of boxes.
+\item[3.] Control constructs:\\
+     if-then-else statements testing relations between integers,
+     dimensions, boxes, or boolean variables.
+\item[4.] Modularization constructs:\\
+     macros with up to 9~parameters (can be viewed as procedures without
+     the concept of local variables).
+\end{enumerate}
+
+Although the programming
+facilities of \TeX{} hardly exceed the abilities of a Turing machine,
+they are sufficient to
+handle small programs. How about the graphics facilities?
+Although \TeX{} has no built-in graphics facilities, it
+allows the placement of characters in arbitrary positions on
+the page. Therefore, complex pictures can be synthesized from elementary
+picture elements treated as characters. Lamport has included such
+a picture drawing environment in his macro package \LaTeX, using
+quarter circles of different sizes and line segments (with and without
+arrow heads) of different slopes as basic elements; see~\cite{LaTeX}.
+These elements are sufficient for drawing trees.
+
+This survey of \TeX's capabilities implies that \TeX{} may be a suitable
+text processing system to implement a tree drawing algorithm directly.
+We base our algorithm on the RT~algorithm, because this algorithm
+gives, aesthetically, the most pleasing results. In the first version
+presented here, we
+restrict ourselves to unary-binary trees, although our method is
+applicable to arbitrary multiway trees. But to take advantage
+of the text processing environment, we expand the algorithm to allow
+labeled nodes.
+
+In contrast to previous tree drawing programs, we feel no necessity to
+position the nodes of a tree on a fixed grid. While this may be
+reasonable for a plotter with a coarse resolution, it is certainly not
+necessary for \TeX, a system that is capable of handling
+arbitrary dimensions
+and producing device \emph{independent} output.
+
+
+\section{A representation method for \TeX{}trees}
+
+The first problem to be solved in implementing our tree drawing algorithm
+is how to choose a good internal representation
+for trees. A straightforward adaptation
+of the implementation by Reingold and Tilford requires, for each node,
+at least:
+%
+\begin{enumerate}
+\item two pointers to the children of the node,
+\item two dimensions for the offset to the left and the right child (these
+      may be different once there are labels of different widths to the
+      left and right of the nodes),
+\item two dimensions for the $x$- and $y$-coordinates of the final
+      position of the nodes,
+\item three or four labels, and
+\item one token to store the geometric shape (circle, square, framed text, etc.)
+      of the node.
+\end{enumerate}
+%
+Because these data are used frequently in calculations, they should be
+stored in registers (that's what variables are called in \TeX)
+rather than being recomputed, to obtain
+reasonably fast performance. This gives a total of $10N$ registers for
+a tree with $N$ nodes, which quickly exceeds
+\TeX's limited supply of registers. Therefore, we present a
+modified algorithm hand-tailored to the abilities of \TeX{}.
+We start with the following observation.
+Suppose a unary-binary tree is built bottom-up, using a postorder
+traversal. This can be done by repeating the following three steps in
+an order determined by the tree to be built.
+
+\begin{enumerate}
+\item Create a new subtree consisting of one external node.
+\item Create a new subtree by appending the two subtrees last created 
+      to a new binary node; see \fig{Construct}.
+\item Create a new subtree by appending the subtree created last as a left,
+      right, or unary subtree of a new node; see \fig{Construct}.
+\end{enumerate}
+
+(A pointer to) each subtree that has been
+created in steps 1--3 is pushed onto a stack, and
+steps 2 and 3 remove two trees or one tree, respectively,
+from the stack before the push
+operation is carried out. The tree to be built is
+the tree remaining on the stack.
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\treesymbol{\lvls{2}}%
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}
+$+$
+\treesymbol{\lvls{2}}%
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad
+$\Longrightarrow$\quad
+\treesymbol{\lvls{2}}%
+\treesymbol{\lvls{2}}%
+\node{\type{dot}}%
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%
+\end{Tree}
+
+\vskip\baselineskip
+
+\begin{Tree}
+\treesymbol{\lvls{2}}%
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad
+$\Longrightarrow$\quad
+\treesymbol{\lvls{2}}%
+\node{\leftonly\type{dot}}%
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%
+\quad or\quad
+\treesymbol{\lvls{2}}%
+\node{\unary\type{dot}}%
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%
+\quad or\quad
+\treesymbol{\lvls{2}}%
+\node{\rightonly\type{dot}}%
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%
+\end{Tree}
+
+\caption{Construction steps 2 and 3}
+\label{Construct}
+\end{Figure}
+
+This tree traversal is performed twice in the RT~algorithm.
+During the first pass,
+at each execution of steps 2 or~3, the relative positions of the
+subtree(s) and of the new node are computed.
+A closer examination of the RT~algorithm reveals that information about the
+subtree's coordinates is not needed during this pass; the contour information
+alone is sufficient. Complete information is only needed in the second
+traversal, when the tree is really drawn. This is where we can use
+a special feature of \TeX{} that allows us to save registers.
+Unlike Pascal, \TeX{} has the capability of
+storing a drawing in a single box register that can be positioned freely in
+later drawings. This means that in our implementation the two passes
+of the original RT~algorithm can be woven into a single pass,
+storing the contour and drawing of each subtree on the stack.
+Although the latter is a complex object, it takes only one of
+\TeX's precious registers.
+
+
+\section{The internal representation}
+
+Given a tree, the corresponding \TeX{}tree is a box containing
+the ``drawing'' of the tree, together with some additional
+information about the contour of the tree.
+The reference point of a \TeX{}tree-box is always in the root of the
+tree. The height, depth, and width of the box of a \TeX{}tree are
+of no importance in this context.
+
+The additional information about the contour of the tree is stored in some
+registers for numbers and dimensions and
+is needed in order to put subtrees together to form a larger tree.
+An array \var{loff} of dimensions contains for each
+level of the tree the horizontal offset between the
+left end of the leftmost node at the current level and the
+left end of the leftmost node at the next level.
+The horizontal offset between the root
+and the leftmost node of the whole tree is hold in \var{lmoff}, and 
+the horizontal offset between the root and the leftmost node at
+the bottom level of the tree is hold in \var{lboff}.
+Finally, \var{ltop} holds the distance between the reference point
+of the tree and the leftmost end of the root.
+We use
+\var{roff}, \var{rmoff}, \var{rboff}, and \var{rtop}
+as the corresponding variables for ``left'' replaced by ``right.'' Finally,
+\var{height} holds the height of the tree, and \var{type} holds the
+geometric shape of the root of the tree. \fig{TeXtree} shows an example 
+\TeX{}tree, that is a tree drawing and the corresponding additional information.
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\e\ir\ir\e
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}%
+  \i % A
+\end{Tree}
+\leavevmode
+\stack{-10pt}{\vd@st}{%
+     -10pt\\10pt\\10pt\\\var{loff}}%
+\hspace{1em}%
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%
+\hspace{1em}%
+\stack{-10pt}{\vd@st}{%
+     15pt\\5pt\\-10pt\\\var{roff}}%
+
+\vskip\baselineskip\raggedright
+height:~3, type:~dot, ltop:~2pt, rtop:~2pt, lmoff:~-10pt, rmoff:~20pt, lboff:~10pt,
+rboff:~10pt.
+
+\caption{A \TeX{}tree consists of the drawing of the tree and the additional 
+information. The width of the dots is 4pt, the minimal separation between
+adjacent nodes is 16pt, making for a distance of 20pt center to center.
+The length of the small rule labeling 
+one of the nodes is 5pt. The column left (right)
+of the tree drawing is the array \var{loff} (\var{roff}),
+describing the left (right) contour of the tree. At each level,
+the dimension given is the horizontal
+offset between the border at the current and 
+at the next level. The offset between
+the left border of the root node and the leftmost node at level~1 is -10pt,
+the offset between the right border of the root node and the rightmost node at
+level~1 is 15pt, etc.}
+\label{TeXtree}
+\end{Figure}
+
+Given two \TeX{}trees \var{A} and \var{B},
+how can a new \TeX{}tree \var{C} be built that
+consists of a new root and has \var{A} and \var{B} as subtrees?
+An example is given in \fig{AddInfo}.
+First we determine which tree is higher; this is
+\var{B} in the example.
+Then we have to compute the minimal distance
+between the roots of \var{A} and \var{B}, such that at all levels
+of the trees there is free space of at least \var{minsep} between
+the trees when they are drawn side by side.
+For this purpose we keep track of two values, \var{totsep} and
+\var{currsep}. The variables \var{totsep} and \var{currsep}
+hold the total distance between the roots and the distance
+between the rightmost node of \var{A} and the leftmost node
+of \var{B} at the current level. To calculate
+\var{totsep} and \var{currsep}, we start at level 0 and
+visit each level of the trees until we reach the bottommost level
+of the smaller tree; this is \var{A} in our example.
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\e\ir\ir\e
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}%
+  \i % A
+\end{Tree}
+\leavevmode
+A: \stack{-10pt}{\vd@st}{%
+     -10pt\\10pt\\10pt\\\ \\\var{loff}(\var{A})}%
+\hspace{1em}%
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%
+\hspace{1em}%
+\stack{-10pt}{\vd@st}{%
+     15pt\\5pt\\-10pt\\\ \\\var{roff}(\var{A})}%
+\qquad
+\begin{Tree}
+\e\il\e\i\il\il\ir % B
+\end{Tree}
+\leavevmode
+B: \stack{-10pt}{\vd@st}{%
+     10pt\\-10pt\\-10pt\\-10pt\\-10pt\\\ \\\var{loff}(\var{B})}%
+\hspace{1em}%
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%
+\hspace{1em}%
+\stack{-10pt}{\vd@st}{%
+     10pt\\-10pt\\-10pt\\10pt\\-30pt\\\ \\\var{roff}(\var{B})}%
+\\[\figspace]
+\begin{Tree}
+\e\ir\ir\e
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}%
+  \i % A
+\e\il\e\i\il\il\ir % B
+\i % C
+\end{Tree}
+\leavevmode
+C: \stack{-10pt}{\vd@st}{%
+     -20\\-10pt\\%
+     \makebox[0pt][r]{\var{loff}(\var{A})$\smash{\left\{\vrule height\vd@st
+          depth\vd@st width0pt\right.}$ }%
+     10pt\\10pt\\%
+     \makebox[0pt][r]{$\longrightarrow$ }%
+     10pt\\%
+     \makebox[0pt][r]{\raisebox{-.5\vd@st}{\var{loff}(\var{B})$\smash
+          {\left\{\vrule height.5\vd@st
+          depth.5\vd@st width0pt\right.}$ }}%
+     \makebox[0pt][r]{-}10pt\\\ \\\var{loff}(\var{C})}%
+\hspace{1em}%
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%
+\hspace{1em}%
+\stack{-10pt}{\vd@st}{%
+     20pt\\10pt\\-10pt\\-10pt%
+     \makebox[0pt][l]{\raisebox{-.5\vd@st}{
+          $\smash{\left\}\vrule height2.5\vd@st
+          depth2.5\vd@st width0pt\right.}$\var{roff}(\var{B})}}%
+     \\10pt\\-30pt\\\ \\\var{roff}(\var{C})}%
+
+\vspace{\figspace}
+\centering
+\begin{tabular}{|l|r|r|r|}
+\hline
+&\multic{1}{c|}{\var{A}}&\multic{1}{c|}{\var{B}}&\multic{1}{c|}{\var{C}}\\
+\hline
+height&\multic{1}{c|}{3}&  \multic{1}{c|}{5}&  \multic{1}{c|}{6}\\
+type&  \multic{1}{c|}{dot}&\multic{1}{c|}{dot}&\multic{1}{c|}{dot}\\
+ltop&  2pt&                2pt&                2pt\\
+rtop&  2pt&                2pt&                2pt\\
+lmoff& -10pt&              -30pt&              -30pt\\
+rmoff& 20pt&               10pt&               30pt\\
+lboff& 10pt&               -30pt&              -10pt\\
+rboff& 10pt&               -30pt&              -10pt\\
+\hline
+\end{tabular}\qquad
+\begin{tabular}{|c|r|r|}
+\hline
+\multic{1}{|c|}{level}&\multic{1}{c|}{\var{totsep}}&
+                      \multic{1}{c|}{\var{currsep}}\\
+\hline
+0&20pt&0/16pt\\
+1&25pt&11/16pt\\
+2&40pt&1/16pt\\
+3&40pt&16pt\\
+\hline
+\end{tabular}
+\caption{The \TeX{}trees \var{A} and~\var{B} are combined to form the
+larger \TeX{}\-tree~\var{C}. The first table gives the additional
+information of the three \TeX{}trees,
+and the second table gives the
+history of the computation for \var{totsep} and \var{currsep}.}
+\label{AddInfo}
+\end{Figure}
+
+At level 0, the distance between the roots of \var{A} and \var{B}
+should be at least \var{minsep}. Therefore, we set
+$\var{totsep}:=\var{minsep} + \var{rtop}(\var{A})
++ \var{ltop}(\var{B})$ and $\var{currsep}:=\var{minsep}$.
+Using $\var{roff}(\var{A})$ and $\var{loff}(\var{B})$, we can
+calculate \var{currsep} for the next level.
+If $\var{currsep} < \var{minsep}$, we have to increase \var{totsep} by
+the difference and update \var{currsep}. This process is
+repeated until we reach the lowest level of \var{A}
+at which point \var{totsep} holds the final distance between the
+nodes of \var{A} and \var{B}, as calculated by the RT~algorithm.
+If the root of \var{C} is a significant node, then the additional space,
+which is 0pt by default, is added to \var{totsep}.
+However, the approach of synthesizing
+drawings from simple graphics characters allows only a finite
+number of orientations for the tree edges; therefore, \var{totsep}
+must be increased slightly to fit the next orientation
+available.
+
+Now we are ready to build the box of \TeX{}tree~\var{C}.
+Simply put \var{A} and~\var{B} side by side, with the reference
+points \var{totsep}~units apart, insert a new node
+above them, and connect the parent and children by edges.
+Next, we compute the additional information
+for \var{C}. This can be done by using the additional information
+for \var{A} and~\var{B}.
+Note that most components of $\var{roff}(\var{C})$ and
+$\var{lroff}(\var{C})$ are the same as in the higher tree, which
+is \var{B} in our case.
+So, if we can avoid moving this information around, 
+the number of counters we have to access to update the additional information
+for \var{C} is within a small constant of the height of~\var{A}.
+Hence, we can apply the same argument as
+in~\cite{TidierTrees}, which gives
+us a running time of $\O(N)$ for drawing a tree with N nodes.
+
+We must design the allocation of storage registers for
+the additional information of \TeX{}trees carefully to fulfill the
+following requirement. If a new tree is built from
+two subtrees, the additional information of the new tree
+shares storage with its larger subtree.
+Organizational overhead, that is,
+pointers that keep track of the locations of different parts of additional
+information, must be avoided.
+This means that the additional information
+for one \TeX{}tree should be stored in a sequence
+of consecutive dimension registers
+such that only one pointer for access to the first element
+in this sequence is needed. On the other hand, each parent
+tree is higher and, therefore, needs more storage than its subtrees.
+So we must ensure that there is always enough space in the sequence
+for more information.
+
+The obvious way to fulfill these requirements is to use a stack and to
+allow only the topmost \TeX{}trees of this stack to be
+combined into a larger tree at any time.
+This leads to the following allocation of registers: A contiguous sequence of
+box registers contains the treeboxes of the subtrees in the stack. A
+contiguous sequence of token registers contains the type information for the
+nodes of the subtrees in the stack. For each subtree in the stack,
+a contiguous sequence of dimension registers contains the contour
+information of the subtree. The ordering of these groups of dimension
+registers reflects the ordering of the subtrees in the
+stack. Finally, a contiguous sequence of counter registers contains
+the height and the address of the first dimension register for
+each subtree in the stack. Four address counters store the addresses
+of the last treebox, type information, height, and address of contour
+information. A sketch of the register organization for a stack of \TeX{}trees
+is provided in \fig{Registers}.
+
+\begin{Figure}
+Dimension registers\\
+\var{lmoff}(1) \var{rmoff}(1) \var{lboff}(1) \var{rboff}(1) \var{ltop}(1)
+               \var{rtop}(1)\\
+\var{loff}($h_1$) \var{roff}($h_1$) \dots\ \var{loff}(1) \var{roff}(1)\\
+\dots\\
+\var{lmoff}($n$) \var{rmoff}($n$) \var{lboff}($n$) \var{rboff}($n$)
+                 \var{ltop}($n$) \var{rtop}($n$)\\
+\var{loff}($h_n$) \var{roff}($h_n$) \dots\ \var{loff}(1) \var{roff}(1)\\
+\mbox{}\\
+Counter registers\\
+\var{lasttreebox} \var{lasttreeheight} \var{lasttreeinfo} \var{lasttreetype}\\
+\var{treeheight}(1) \var{diminfo}(1) \dots\ \var{treeheight}($n$)
+                    \var{diminfo}($n$)\\
+\mbox{}\\
+Box registers\\
+\var{treebox}(1) \dots\ \var{treebox}($n$)\\
+\mbox{}\\
+Token registers\\
+\var{type}(1) \dots\ \var{type}($n$)
+
+\caption{\var{lasttreebox}, \var{lasttreeheight}, \var{lasttreeinfo},
+\var{lasttreetype} contain pointers to \var{treebox}($n$)
+\var{treeheight}($n$), \var{lmoff}($n$), \var{type}($n$),
+\var{diminfo}($i$) contains a pointer to
+\var{lmoff}($i$). Unused dimension registers are
+allowed between the dimension registers of subsequent trees. The counter
+registers \var{lasttreebox},\ldots,\var{diminfo}($n$) serve as a directory
+mechanism to access the \TeX{}trees on the stack.}
+\label{Registers}
+\end{Figure}
+
+
+When a new node is pushed onto the stack, the treebox, type information,
+height, address of contour information, and contour information are
+stored in the next free registers of the appropriate type, and the
+four address counters are updated accordingly.
+
+When a new tree is formed from the topmost subtrees on the stack,
+the treebox, type information, height, and address of contour information
+of the new tree are sorted in the registers formerly used by the bottommost
+subtree that has occurred in the construction step, 
+and the four address registers are
+updated accordingly. This means that this information for the subtrees
+is no longer accessible. The contour information of the new subtree
+is stored in the same registers as the contour information of the larger
+subtree used in the construction, apart from the left and right offset
+of the root to the left and right child, which are stored in the
+following dimension registers. This means that gaps can occur
+between the contour information of subtrees in the
+stack, namely when the right subtree, which is in a higher position in the
+stack, is higher than the left one. To avoid these
+gaps, the user can specify an option \verb.\lefttop. when entering a
+binary node, which makes the topmost tree in the stack the
+left subtree of the node.
+
+This stack concept also has consequences for the design of the user interface
+that is discussed in Section~\ref{Interface}.
+
+\section{Space cost analysis}
+
+Suppose we want to draw a unary-binary tree $T$ of height $h$ having
+$N$ nodes\footnote{The height $h$ and the number of nodes $N$ refer to the
+drawing of the tree. $N$ is the number of circles, squares,~etc., actually
+drawn, and $h$ is the number of levels in the drawing minus 1.}.
+According to our internal representation,
+for each subtree in the stack we need:
+
+\begin{enumerate}
+\item one box register to store the box of the \TeX{}tree;
+\item one token register to store the type of the root of the subtree;
+\item $2h^\prime+6$ dimension registers to store the additional
+      information, where $h^\prime$ is the height of the
+      subtree; and
+\item three counter registers to store the register numbers of the
+      box register, the token register, and the first dimension register above.
+\end{enumerate}
+
+\begin{lemma}
+Let $T$ be a unary-binary tree of height~$h$ and size~$N$; then:
+\begin{enumerate}
+\item at any time, there are at most $h+1$ subtrees of $T$ on the
+      stack; and
+\item for each set $\T$ of subtrees of $T$ that are on the stack
+      simultaneously we have
+      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1) \le N$$
+\end{enumerate}
+\end{lemma}
+
+The lemma implies that our implementation 
+uses at most $9h+2N$~registers. 
+To compare this with the
+$10N$ registers used in the straightforward implementation,
+an estimation of the average height of a tree with $N$ nodes is
+needed. Several results, depending on the type of trees and of the
+randomization model, are cited in \fig{Stat}, which
+compares the number of registers used in a straightforward
+implementation with the average number of registers used in our
+implementation. This table shows clearly the advantage of our
+implementation.
+
+\begin{Figure}
+\centering
+
+\begin{tabular}{|c|c|c|c|c|}
+\hline
+&registers&\multicolumn{3}{c|}{average registers}\\
+\cline{3-5}
+nodes&(straight-&&unary-binary&binary\\
+&forward)&binary trees&trees&search trees\\
+$N$&&($2\sqrt{\pi N}$) \cite{BinaryTrees}&
+     ($\sqrt{3\pi N}$) ~\cite{BinaryTrees}&
+     ($4.311\log N$) \cite{BinarySearchTrees}\\
+\hline
+\ds10 & \ds100 & 120.89 & 107.37 & 109.34 \\
+\ds20 & \ds200 & 182.68 & 163.56 & 156.23 \\
+\ds30 & \ds300 & 234.75 & 211.33 & 191.96 \\
+\ds40 & \ds400 & 281.78 & 254.75 & 223.12 \\
+\ds50 & \ds500 & 325.60 & 295.37 & 251.78 \\
+\ds60 & \ds600 & 367.13 & 334.02 & 278.86 \\
+\ds70 & \ds700 & 406.93 & 371.17 & 304.84 \\
+\ds80 & \ds800 & 445.36 & 407.13 & 330.02 \\
+\ds90 & \ds900 & 482.67 & 442.12 & 354.59 \\
+100 & 1000 & 519.04 & 476.30 & 378.68 \\
+\hline
+\end{tabular}
+
+\caption{The numbers of registers used by a straightforward implementation
+(second column) and by our modified implementation (third to fifth column)
+of the RT~algorithm are
+given for different types of trees and randomization models.
+The formulas in parentheses indicate the average height of the respective 
+classes of trees.}
+\label{Stat}
+\end{Figure}
+
+
+\section{The user interface}\label{Interface}
+
+The user interface of \TreeTeX{} has been designed in the spirit of
+the thorough separation of the logical description of document components
+and their layout; see~\cite{DocumentFormatting,GML}. This concept
+ensures both uniformity and flexibility of document layout and frees
+authors from layout problems that have nothing to do with the
+substance of their work. For some powerful implementations and projects
+see \cite{Tables,Karlsruhe,LaTeX,Grif,Scribe}.
+
+The description of a tree consists of a description of its nodes
+in postorder. Each description of a node, in turn, has to specify
+the outdegree, the geometric shape and the labels of the node.
+Defaults are provided for all specifications,
+thereby allowing the user to omit many definitions
+if the defaults match what he or she wants.
+
+A separate style command defines layout parameters for tree drawings
+that are valid for all trees of a document.
+Layout parameters include the font to be used for labels, the diameter
+of circle nodes, the vertical distance between two subsequent levels
+of the tree, and the minimal horizontal distance between nodes.
+
+Standard versions of \TeX{} provide only a limited number of
+font and circle sizes. Hence, the user of the style command must make
+sure that the specified sizes can be realized. This is especially
+cumbersome when everything has to be magnified for later reproduction
+with reduction. But the style variables can be made parametric for
+installations that provide scalable fonts and replace \LaTeX{}'s
+circle- and line-drawing commands with routines that provide arbitrary
+diameters and slopes.
+
+Three examples of tree descriptions are given in 
+Figures~\ref{firstex}--\ref{lastex}.
+A more detailed description of the user interface is 
+given in~\cite{Exeter}.
+
+\section{Conclusions}
+
+We hope that, by now, we have convinced the reader of the main advantages
+of \TreeTeX{}: It integrates graphics and text; it is portable to all
+sites running \TeX{}; 
+and it is easy to use for the author, because it derives the drawing 
+of a tree from a purely structural description. But our decision to
+implement \TreeTeX{} as a \TeX{} macro package has also some
+drawbacks, both for the programmer and for the user of the system.
+
+>From the programmer's point of view, \TeX{}'s macro language is
+a low level programming language. Hence, maintaining and extending
+the package is a more tedious task than it would be if we had used
+a higher level language with better support for modularization.
+
+>From the author's point of view, \TreeTeX{}'s limitations lie in
+speed, size of trees, and graphical primitives.
+Typesetting all the trees in this article takes about two~minutes on
+a VAX~750, and typesetting a complete binary tree with 63~internal
+and 64~external nodes takes about one~minute on the same machine.
+The size of the trees is limited by three factors, namely,
+the number of registers, the complexity of the nested boxes that
+contain the drawing of a tree, and the limited number of slopes
+that are available for the edges, the latter being the most severe 
+problem at present. Hence, the main area of application for
+\TreeTeX{} is modest use such as in textbooks; displaying
+large amounts of statistical data, for example, is out of the question.
+
+Currently edges and circular nodes are drawn from \LaTeX{}'s set of 
+predefined graphical characters. Hence, \TreeTeX{} cannot draw
+arbitrarily wide trees or large circular nodes. We consider
+this restriction, however, to be a temporary one, since a committee inside
+the \TeX{} Users Group is  working on standard graphic 
+extensions to \TeX{} that will remove these limitations.
+
+As to further developments of \TreeTeX{}, it would be desirable to
+draw larger classes of trees, for example multiway trees, and to allow
+labels not only for nodes, but also for edges and whole subtrees.
+
+
+\Treestyle{\vdist{60pt}}
+\dummyhalfcenterdim@n=10pt
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\node{\external\bnth{first}\cntr{1}\lft{Beeton}}
+\node{\external\cntr{3}\rght{Kellermann}}
+\node{\cntr{2}\lft{Carnes}}
+\node{\external\cntr{6}\lft{Plass}}
+\node{\external\bnth{last}\cntr{8}\rght{Tobin}}
+\node{\cntr{7}\rght{Spivak}}
+\node{\leftonly\cntr{5}\rght{Lamport}}
+\node{\cntr{4}\rght{Knuth}}
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\
+
+\begin{verbatim}
+\begin{Tree}
+\node{\external\bnth{first}\cntr{1}\lft{Beeton}}
+\node{\external\cntr{3}\rght{Kellermann}}
+\node{\cntr{2}\lft{Carnes}}
+\node{\external\cntr{6}\lft{Plass}}
+\node{\external\bnth{last}\cntr{8}\rght{Tobin}}
+\node{\cntr{7}\rght{Spivak}}
+\node{\leftonly\cntr{5}\rght{Lamport}}
+\node{\cntr{4}\rght{Knuth}}
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim}
+
+\caption{This is an example of a tree that includes labels.}
+\label{firstex}
+\end{Figure}
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\node{\external\type{frame}\bnth{first}\cntr{Beeton}}
+\node{\external\type{frame}\cntr{Kellermann}}
+\node{\type{frame}\cntr{Carnes}}
+\node{\external\type{frame}\cntr{Plass}}
+\node{\external\type{frame}\bnth{last}\cntr{Tobin}}
+\node{\type{frame}\cntr{Spivak}}
+\node{\leftonly\type{frame}\cntr{Lamport}}
+\node{\type{frame}\cntr{Knuth}}
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\
+
+\begin{verbatim}
+\begin{Tree}
+\node{\external\type{frame}\bnth{first}\cntr{Beeton}}
+\node{\external\type{frame}\cntr{Kellermann}}
+\node{\type{frame}\cntr{Carnes}}
+\node{\external\type{frame}\cntr{Plass}}
+\node{\external\type{frame}\bnth{last}\cntr{Tobin}}
+\node{\type{frame}\cntr{Spivak}}
+\node{\leftonly\type{frame}\cntr{Lamport}}
+\node{\type{frame}\cntr{Knuth}}
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim}
+\caption{This is an example of a tree with framed center labels.}
+\end{Figure}
+
+\begin{Figure}
+\Treestyle{\treefonts{\small\it}\nodesize{16pt}\vdist{40pt}\minsep{16pt}}
+\centering
+\begin{Tree}
+\node{\external\bnth{first}\cntr{1}\lft{Beeton}}
+\node{\external\cntr{3}\rght{Kellermann}}
+\node{\cntr{2}\lft{Carnes}}
+\node{\external\cntr{6}\lft{Plass}}
+\node{\external\bnth{last}\cntr{8}\rght{Tobin}}
+\node{\cntr{7}\rght{Spivak}}
+\node{\leftonly\cntr{5}\rght{Lamport}}
+\node{\cntr{4}\rght{Knuth}}
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\
+
+\caption{This tree was produced from the same logical description as in
+Figure~\ref{firstex}, but with different style parameters}
+\label{lastex}
+\end{Figure}
+
+\clearpage
+\bibliography{trees}
+\end{document}
+
+
diff --git a/Master/texmf-dist/doc/plain/treetex/readme b/Master/texmf-dist/doc/plain/treetex/readme
new file mode 100644
index 00000000000..575afc2fe0b
--- /dev/null
+++ b/Master/texmf-dist/doc/plain/treetex/readme
@@ -0,0 +1,19 @@
+TreeTeX is a public domain macro package for drawing
+trees with TeX. It may be freely distributed, provided
+that the following files are kept together:
+
+  README epodd.bbl epodd.tex tree_doc.bbl tree_doc.tex
+  classes.tex l_pic.tex treetex.tex
+
+Copyright is with Anne Brueggemann-Klein and Derick Wood.
+
+All remarks, bug reports etc. should be directed to
+
+  Dr. Anne Brueggemann-Klein
+  Institut fuer Informatik
+  Rheinstr. 10--12
+  7800 Freiburg, West Germany
+
+  email: abk@sun1.ruf.uni-freiburg.dbp.de
+
+
diff --git a/Master/texmf-dist/doc/plain/treetex/tree_doc.bbl b/Master/texmf-dist/doc/plain/treetex/tree_doc.bbl
new file mode 100644
index 00000000000..88c540ee0b7
--- /dev/null
+++ b/Master/texmf-dist/doc/plain/treetex/tree_doc.bbl
@@ -0,0 +1,97 @@
+\begin{thebibliography}{10}                                                     
+                                                                                
+\bibitem{Tables}                                                                
+R.~J. Beach.                                                                    
+\newblock {\it Setting Tables and Illustrations with Style}.                    
+\newblock PhD thesis, University of Waterloo, 1985.                             
+                                                                                
+\bibitem{Karlsruhe}                                                             
+A. Br{\"u}ggemann-Klein, P. Dolland, and A. Heinz.                              
+\newblock How to please authors and publishers: a versatile document            
+  preparation system at {K}arlsruhe.                                            
+\newblock In J. D\'esarm\'enien, editor, {\it {\TeX{}} for Scientific           
+  Documentation}, Strasbourg, France, June 1986.                                
+\newblock LNCS~236.                                                             
+                                                                                
+\bibitem{TreeTeX}                                                               
+A. Br{\"u}ggemann-Klein and D. Wood.                                            
+\newblock {\it Tree{\TeX}: Documentation and User Handbook}.                    
+\newblock Technical Report, University of Waterloo, 1987.                       
+                                                                                
+\bibitem{AverageHeight}                                                         
+N.G. de~Bruijn, D. Knuth, and S.O. Rice.                                        
+\newblock The average height of planted plane trees.                            
+\newblock In R.C. Read, editor, {\it Graph Theory and Computing}, 1972.         
+                                                                                
+\bibitem{BinarySearchTrees}                                                     
+L. Devroye.                                                                     
+\newblock A note on the height of binary search trees.                          
+\newblock {\it Journal of the ACM}, 33(3), July 1986.                           
+                                                                                
+\bibitem{BinaryTrees}                                                           
+Ph. Flajolet and A. Odlyzko.                                                    
+\newblock The average height of binary trees and other simple trees.            
+\newblock {\it Journal of Computer and System Sciences}, 25, 1982.              
+                                                                                
+\bibitem{DocumentFormatting}                                                    
+R. Furuta, J. Scofield, and A. Shaw.                                            
+\newblock Document formatting systems: surveys, concepts, issues.               
+\newblock {\it Computing Surveys}, 14(3), 1982.                                 
+                                                                                
+\bibitem{GML}                                                                   
+Ch.~F. Goldfarb.                                                                
+\newblock A generalized approach to document markup.                            
+\newblock {\it SIGPLAN Notices of the ACM}, June 1981.                          
+                                                                                
+\bibitem{ACP}                                                                   
+D.~E. Knuth.                                                                    
+\newblock {\it Fundamental Algorithms}.                                         
+\newblock Volume~1 of {\it The Art of Computer Programming}, Addison-Wesley,    
+  Reading, Massachusetts, 1973.                                                 
+                                                                                
+\bibitem{TeXbook}                                                               
+D.~E. Knuth.                                                                    
+\newblock {\it The {\TeX{}}book}.                                               
+\newblock Volume~A of {\it Computers \& Typesetting}, Addison-Wesley, Reading,  
+  Massachusetts, 1986.                                                          
+                                                                                
+\bibitem{LaTeX}                                                                 
+L. Lamport.                                                                     
+\newblock {\it {\LaTeX}, User's Guide \& Reference Manual}.                     
+\newblock Addison-Wesley, Reading, Massachusetts, 1986.
+ 
+\bibitem{Brother}
+Th. Ottmann, H.-W. Six.
+\newblock Eine neue Klasse ausgeglichener Bin\"arb\"aume.
+\newblock {\it Angewandte Informatik}, 9, 1976.                         
+                                                                                
+\bibitem{Grif}                                                                  
+V. Quint, I. Vatton, and H. Bedor.                                              
+\newblock Grif: an interactive environment for {\TeX{}}.                        
+\newblock In J. D\'esarm\'enien, editor, {\it {\TeX{}} for Scientific           
+  Documentation}, Strasbourg, France, June 1986.                                
+\newblock LNCS~236.                                                             
+                                                                                
+\bibitem{Scribe}                                                                
+B.~K. Reid.                                                                     
+\newblock {\it Scribe: A Document Specification Language and its Compiler}.     
+\newblock PhD thesis, Carnegie Mellon University, 1980.                         
+                                                                                
+\bibitem{TidierTrees}                                                           
+E.~M. Reingold and J.~S. Tilford.                                               
+\newblock Tidier drawings of tree.                                              
+\newblock {\it IEEE Transactions on Software Engineering}, 7(2), March 1981.    
+                                                                                
+\bibitem{Complexity}                                                            
+K.~J. Supowit and E.~M. Reingold.                                               
+\newblock The complexity of drawing trees nicely.                               
+\newblock {\it Acta Informatica}, 18, 1983.                                     
+                                                                                
+\bibitem{TidyTrees}                                                             
+Ch. Wetherell and A. Shannon.                                                   
+\newblock Tidy drawings of trees.                                               
+\newblock {\it IEEE Transactions on Software Engineering}, 5(5), September      
+  1979.                                                                         
+                                                                                
+\end{thebibliography}                                                           
+                                                                                
diff --git a/Master/texmf-dist/doc/plain/treetex/tree_doc.tex b/Master/texmf-dist/doc/plain/treetex/tree_doc.tex
new file mode 100644
index 00000000000..635d0dab5d6
--- /dev/null
+++ b/Master/texmf-dist/doc/plain/treetex/tree_doc.tex
@@ -0,0 +1,1331 @@
+% This is tree_doc.tex, the documentation for the treetex macro package
+% as it will appear in the conference proceedings of the third European
+% TeX meeting in Exeter, England, 1988.
+
+\documentstyle[12pt,a4]{article} 
+
+\advance\voffset by -2cm                                           
+                                                                                
+\clubpenalty=10000                                                              
+\widowpenalty=10000                                                             
+\def\addcontentsline#1#2#3{\relax}% Some captions are too long for some
+     % TeX installations (buffer size too small)
+
+                                                                                
+\newenvironment{lemma}{\begingroup\samepage\begin{lemmma}\ }{\end{lemmma}%      
+     \endgroup}                                                                 
+\newtheorem{lemmma}{Lemma}[section]                                             
+\newenvironment{proof}{\begin{prooof}\rm\ \nopagebreak}{\end{prooof}}           
+\newcommand{\proofend}{\qquad\ifmmode\Box\else$\Box$\fi}                        
+\newtheorem{prooof}{Proof}                                                      
+\renewcommand{\theprooof}{}   % makes shure that prooof doesn't get numbers       
+\newenvironment{Figure}{\begin{figure}\vspace{1\baselineskip}}%                 
+                       {\vspace{1\baselineskip}\end{figure}}                    
+\newlength{\figspace}         % space between figures in a single               
+\setlength{\figspace}{30pt}   % Figure environment                              
+                                                                                
+\newcommand{\var}[1]{{\it #1\/}}     % use it for names of variables            
+\renewcommand{\emph}[1]{{\em #1\/}}    % use it for emphazided text               
+                                     % (This notion sticks to the               
+                                     % applicative style of markup.)            
+\renewcommand{\O}{{\rm O}}           % O-notation, also for math mode           
+\newcommand{\T}{{\cal T}}            % the set T in math mode                   
+\newcommand{\TreeTeX}{Tree\TeX}                                                 
+\newcommand{\fig}[1]{Figure~\ref{#1}}                                           
+\let\p\par                                                                      
+                                                                                
+\input treetex                                    
+\Treestyle{\vdist{20pt}\minsep{16pt}}                                                        
+\dummyhalfcenterdim@n=2pt                                                       
+                                                                                
+\def\Node(#1,#2){\put(#1,#2){\circle*{4}}}                                      
+\def\Edge(#1,#2,#3,#4,#5){\put(#1,#2){\line(#3,#4){#5}}}                        
+                                                                                
+\def\enode{\node{\external\type{dot}}}                                          
+\def\inode{\node{\type{dot}}}                                                   
+                                                                                
+\def\e{\node{\external\type{dot}}}                                              
+\def\i{\node{\type{dot}}}                                                       
+\def\il{\node{\type{dot}\leftonly}}                                             
+\def\ir{\node{\type{dot}\rightonly}}                                            
+                                                                                
+\newcommand{\stack}[3]{%                                                        
+     \vtop{\settowidth{\hsize}{#1}%                                             
+     \setlength{\leftskip}{0pt plus 1fill}%                                     
+     \setlength{\baselineskip}{#2}#3}}                                          
+                                                                                
+\let\multic\multicolumn                                                         
+                                                                                
+\newlength{\hd} % hidden digit                                                  
+\setbox0\hbox{1}                                                                
+\settowidth{\hd}{\usebox{0}}                                                    
+\newcommand{\ds}{\hspace{\hd}} % digit space                                    
+                                                                                
+\newcommand{\ccol}[1]{\multicolumn{1}{c}{#1}}                                   
+                                                                                
+\hyphenation{post-or-der sym-bol Karls-ruhe bool-ean}                                    
+                                                                                
+\begin{document}                                                                
+                                                                                
+\bibliographystyle{plain}                                                       
+                                                                                
+\title{Drawing Trees Nicely with \TeX\thanks{This work was supported by         
+     a Natural Sciences and Engineering Research Council of Canada              
+     Grant~A-5692 and a Deutsche Forschungsgemeinschaft Grant~Sto167/1-1.
+     It was started during the first author's stay with              
+     the Data Structuring Group in Waterloo.}}                        
+\author{Anne Br\"uggemann-Klein\thanks{Institut f\"ur Informatik,     
+     Universit\"at Freiburg, Rheinstr.~10--12, 7800~Freiburg,
+     West~Germany}\ \and Derick Wood\thanks{Data          
+     Structuring Group, Department of Computer Science, University of           
+     Waterloo, Waterloo, Ontario, N2L~3G1, Canada}}                             
+\maketitle                                                                      
+                                                                                
+\begin{abstract}                                                                
+                                                                                
+Various algorithms have been proposed for the difficult problem of              
+producing aesthetically pleasing drawings of trees, see~%                       
+\cite{TidierTrees,TidyTrees} but                                                
+implementations only exist as ``special purpose software'',                     
+designed for special environments. Therefore,                                   
+many users resort to the                                                        
+drawing facilities available on most personal computers, but the                
+figures obtained in this way still look ``hand-drawn''; their quality is        
+inferior to the quality of the surrounding text that can be realized by         
+today's high quality text processing systems.                                   
+                                                                                
+In this paper we present an entirely new solution that                          
+integrates a tree drawing algorithm into one of the best text                   
+processing systems available. More precisely, we present a \TeX{} macro package 
+\TreeTeX{} that produces a drawing of a tree from a purely logical              
+description. Our approach has three advantages. First, labels           
+for nodes can be handled in a reasonable way. On the one hand, the tree         
+drawing algorithm can compute the widths of the labels and take                 
+them into account for the positioning of the nodes; on the other hand,          
+all the textual parts of the document can be treated uniformly. Second,         
+\TreeTeX{} can be trivially ported to any site running \TeX{}. Finally,         
+modularity in the description of a tree and \TeX{}'s macro capabilities         
+allow for libraries of subtrees and tree classes.                               
+                                                                                
+In addition, we have implemented an option that produces                        
+drawings which make the                                                         
+\emph{structure} of the trees more obvious to the human eye,                    
+even though they may not be as aesthetically pleasing.                          
+                                                                                
+\end{abstract}                                                                  
+                                                                                
+\section{Aesthetical criteria for drawing trees}                                
+                                                                                
+One of the most commonly used data structures in computer science is the tree.  
+As many people are using trees in their research or just as illustration        
+tools, they are usually struggling with the problem of                          
+\emph{drawing} trees. We are concerned primarily with ordered                   
+trees in the sense of~\cite{ACP}, especially binary and unary-binary            
+trees. A binary tree is a finite set of nodes which either                      
+is empty, or consists of a root and two disjoint binary trees called            
+the left and right subtrees of the root. A unary-binary tree is                 
+a finite set of nodes which either is empty, or consists of a root and          
+two disjoint unary-binary trees, or consists of a root and one                  
+nonempty unary-binary tree. An extended binary tree is a binary tree            
+in which each node has either two nonempty subtrees or two                      
+empty subtrees.                                                                 
+                                                                                
+For these trees there                                                           
+are some basic agreements on how they should be drawn, reflecting               
+the top-down and left-right ordering of nodes in a tree;                        
+see \cite{TidierTrees} and \cite{TidyTrees}.                                    
+                                                                                
+\begin{enumerate}                                                               
+\item[1.] Trees impose a distance on the nodes; no node                         
+          should be closer to the root than any of its                          
+          ancestors.                                                            
+\item[2.] Nodes of a tree at the same height should lie on a straight           
+          line, and the straight lines defining the levels should be            
+          parallel.                                                             
+\item[3.] The relative order of nodes on any level should be the same           
+          as in the level order traversal of the tree.                          
+\end{enumerate}                                                                 
+                                                                                
+These axioms guarantee that trees are drawn as planar graphs: edges do          
+not intersect except at nodes. Two further axioms improve the aesthetical       
+appearance of trees:                                                            
+                                                                                
+\begin{enumerate}                                                               
+\item[4.] In a unary-binary tree, each left child should be positioned          
+          to the left of its parent, each                                       
+          right child to the right of its parent, and each unary child          
+          should be positioned below its parent.                                
+\item[5.] A parent should be centered over its children.                        
+\end{enumerate}                                                                 
+                                                                                
+An additional axiom deals with the problem of tree drawings becoming too wide   
+and therefore exceeding the physical limit of the output medium:                
+                                                                                
+\begin{enumerate}                                                               
+\item[6.] Tree drawings should occupy as little width as possible without       
+          violating the other axioms.                                           
+\end{enumerate}                                                                 
+                                                                                
+In \cite{TidyTrees}, Wetherell and Shannon introduce two algorithms for         
+tree drawings, the first of which fulfills axioms~1--5, and the second          
+1--6. However, as Reingold and Tilford in \cite{TidierTrees}                    
+point out, there is a lack of symmetry in the algorithms of                     
+Wetherell and Shannon which may lead to unpleasant results.
+Therefore, Reingold and Tilford introduce a new structured        
+axiom:                                                                          
+                                                                                
+\begin{enumerate}                                                               
+\item[7.] A subtree of a given tree should be                                   
+          drawn the same way regardless of where it occurs in the given tree.   
+\end{enumerate}                                                                 
+                                                                                
+Axiom~7 allows the same tree to be drawn differently when it occurs as          
+a subtree in different trees.                                                   
+Reingold and Tilford give an algorithm which fulfills axioms~1--5               
+and~7. Although                                                                 
+this algorithm doesn't fulfill axiom~6,      
+the aesthetical improvements are well worth the additional space.
+\fig{algorithms} illustrates the benefits of axiom~7, and \fig{narrowtrees}
+shows that the algorithm of Reingold and Tilford violates axiom~6.
+
+\begin{Figure}                                                                  
+\centering
+\leavevmode\noindent                                                                      
+\begin{Tree}
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\node{\external\type{dot}\rght{\unskip\hskip2\mins@p\hskip2\dotw@dth}}
+\enode\enode\inode\enode\enode\inode\inode\inode
+\inode                                                                    
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}                                                               
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\enode
+\enode\enode\inode\enode\enode\inode\inode\inode
+\inode
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\                                                               
+\caption{The left tree is drawn by the algorithm of Wetherell and Shannon,
+and the tidier right one is drawn by the algorithm of Reingold and Tilford.}             
+\label{algorithms}                                                                  
+                                                                                
+\vspace{\figspace}                                                              
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\enode\enode\enode\enode\enode\enode\enode\enode\enode
+\enode\inode\inode\inode                                                                      
+\enode\inode\inode\inode                                                                      
+\enode\inode\inode\inode                                                                      
+\enode\inode\inode\inode 
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\enode\enode\enode\enode\enode\enode\enode\enode
+\node{\external\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\node{\type{dot}\rght{\unskip\hskip\mins@p\hskip\dotw@dth}}
+\enode\inode\inode\inode 
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\caption{The left tree is drawn by the algorithm of Reingold and Tildford, but
+the right tree shows that narrower drawings fulfilling all aesthetic axioms
+are possible.}
+\label{narrowtrees}                                                                  
+\end{Figure}                                                                    
+                                                                                
+                                                                                
+\section{The algorithm of Reingold and Tilford}                                 
+                                                                                
+The algorithm of Reingold and Tilford (hereafter called ``the RT~algorithm'')   
+takes a modular approach to the                                                 
+positioning of nodes: The relative positions of the nodes in a subtree          
+are calculated independently from the rest of the tree. After the               
+relative positions of two subtrees have been calculated, they can be            
+joined as siblings in a larger tree by placing them as close                    
+together as possible and centering the parent node above them.                  
+Incidentally, the modularity principle is the reason that the                   
+algorithm fails to fulfill axiom~6; see~\cite{Complexity}.                      
+Two sibling subtrees are placed as close together as possible,                           
+during a postorder traversal, as follows. At each node \var{T},                 
+imagine that its two subtrees have been drawn and cut out of paper along        
+their contours. Then, starting with the two subtrees superimposed at their      
+roots, move them apart until a minimal agreed upon distance                     
+between the trees is obtained at each level. This can be done gradually:        
+Initially, their roots are separated by some agreed upon minimum                
+distance. Then, at the next lower level,                                        
+they are pushed                                                                 
+apart until the minimum separation is established there.                        
+This process is continued at successively lower levels until the                
+bottom of the shorter subtree is reached. At some levels no movement may be     
+necessary; but at no level are the two subtrees moved closer                    
+together. When the process is complete, the position of the                     
+subtrees is fixed relative to their parent, which is centered over them.        
+Assured that the subtrees will never be placed closer together,                 
+the postorder traversal is continued.                                           
+                                                                                
+A nontrivial implementation of                                                  
+this algorithm has been obtained by Reingold and Tilford that runs              
+in time $\O(N)$, where $N$ is the number of                                     
+nodes of the tree to be drawn.                                                  
+Their crucial idea is to keep track of the contour of the subtrees              
+by special pointers, called threads, such that whenever                         
+two subtrees are joined, only the                                               
+top part of the trees down to the lowest level of the                           
+smaller tree need to be taken into account.                              
+                                                                                
+The RT algorithm is given in \cite{TidierTrees}.
+The nodes are positioned on a fixed grid and are             
+considered to have zero width. No labelling is provided. The algorithm only     
+draws binary trees, but is easily extendable to multiway trees.                 
+                                                                                
+\section{Improving human perception of trees}                                   
+
+It is common understanding in book design that aesthetics and readability
+don't necessarily coincide, and---as Lamport (\cite{LaTeX}) puts it---%
+books are meant to be read, not to be hung on walls. Therefore, readability is
+more important than aesthetics.
+
+When it comes to tree drawings, readability means that the structure of
+a tree must be easily recognizable. This criterion is not always met
+by the RT~algorithm. As an example, there are trees whose structure is very
+different, the only common thing being the fact that they have the same number
+of nodes at each level. The RT~algorithm might assign identical positions to
+these nodes making it very hard to perceive the different structures.
+Hence, we have modified the RT~algorithm such that additional white space
+is inserted between subtrees of 
+\emph{significant} nodes. Here a binary node 
+is called significant if the minimum distance
+between its two subtrees is taken \emph{below} their root level.
+Setting the amount of additional white space to zero retains the original RT~%
+placement. The effect of having nonzero additional white space between 
+the subtrees of significant
+nodes is illustrated in \fig{addspace} .
+
+Another feature we have added to the RT~algorithms is the possibility to draw
+an unextended binary tree with the same placement of nodes as its
+associated extended version. We define the \emph{associated extended version}
+of a binary tree to be the binary tree obtained by replacing each empty subtree
+having a nonempty sibling with a subtree consisting of one node. This feature
+also makes the structure of a tree more prominent; see \fig{extended}.
+
+\begin{Figure}
+\centering
+\leavevmode\noindent                                                                      
+\begin{Tree} 
+\e\il\e\e\i\i\il % the left subtree
+\e\ir\il % the right subtree  
+\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\il\il\il % the left subtree
+\e\e\i\e\i\il % the right subtree
+\i
+\end{Tree}                                                               
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\adds@p10pt
+\begin{Tree}
+\e\il\e\e\i\node{\type{dot}\lft{$\longrightarrow$}}\il % the left subtree
+\e\ir\il % the right subtree  
+\node{\type{dot}\lft{$\longrightarrow$}}
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\il\il\il % the left subtree
+\e\e\i\e\i\il % the right subtree
+\node{\type{dot}\lft{$\longrightarrow$}}
+\end{Tree}                                                               
+\hskip\leftdist\box\TeXTree\hskip\rightdist\
+\adds@p0pt
+
+\caption{The first two trees get the same placement of their nodes
+by the RT~algorithm, although the structure of the two trees is very different.
+The alternative drawings highlight the structure of the trees by adding
+additional white space between the subtrees of 
+($\longrightarrow$) significant nodes.}
+\label{addspace}                                                                                
+\end{Figure} 
+
+\begin{Figure}
+\centering
+\leavevmode\noindent
+\begin{Tree}
+\e\e\i\il\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\e\i\e\i\e\ir\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\extended
+\begin{Tree}
+\e\e\i\il\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\qquad
+\begin{Tree}
+\e\e\i\e\i\e\ir\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\\
+\noextended
+\begin{Tree}
+\e\e\i\e\i\e\e\i\i
+\end{Tree}
+\hskip\leftdist\box\TeXTree\hskip\rightdist\  
+\caption{In the first two drawings, the RT~algorithm assigns the same placement 
+to the nodes of two trees although their structure is very different. The modified
+RT~algorithms highlights the structure of the trees by optionally
+drawing them like their extended
+counterpart, which is given in the second row.}
+\label{extended}
+\end{Figure}
+
+
+\section{Trees in a document preparation environment}                           
+                                                                                
+Drawings of trees usually don't come alone, but are included in some text       
+which is itself typeset by a text processing system. Therefore, a typical      
+scenario is a pipe of three stages. First comes the tree drawing             
+program which calculates the positioning of the nodes of the tree to            
+be drawn and outputs a description of the tree drawing in                       
+some graphics language; next comes a graphics system which transforms this      
+description into an intermediate language which can be interpreted by the output
+device; and finally comes the                                                   
+text processing system which integrates the output of the                       
+graphics system into the text.                                 
+                                                                                
+This scenario loses its linear structure once nodes have to be labelled, since
+the labelling influences the positioning of the nodes. Labels usually occur
+inside, to the left of, to the right of, or beneath nodes (the latter only for
+external nodes), and their extensions certainly should be taken into account
+by the  tree drawing algorithm. But the labels have to be typeset first
+in order to determine their extensions,
+preferably by the typesetting program that
+is used for the regular text, because this method makes for the uniformity in the textual
+parts of the document and provides the author with the full power of the
+text processing system for composing the labels. Hence, a more complex
+communication scheme than a simple pipe is required.
+
+Although a system of two processes running simultaneously might be the most
+elegant solution, we wanted a system that is easily portable to
+a large range of hardware at our sites 
+including personal computers with single process
+operating systems.                                                           
+Therefore, we thought of using a text processing system                              
+having programming facilities powerful enough to program a tree drawing algorithm                      
+and graphics facilities powerful enough                                                             
+to draw a tree. One text processing system                  
+rendering outstanding typographic quality and good enough programming           
+facilities is \TeX, developed by Knuth at Stanford University;                  
+see~\cite{TeXbook}.                                                             
+The \TeX{} system includes the following programming facilities:                
+                                                                                
+\begin{enumerate}                                                               
+\item[1.] datatypes:\\                                                          
+     integers~(256), dimensions\footnote{The term \emph{dimension} is used      
+     in \TeX\ to describe physical measurements of typographical objects,       
+     like the length of a word.}~(512), boxes~(256), tokenlists~(256), boolean  
+     variables~(unrestricted)                                                   
+\item[2.] elementary statements:\\                                              
+     $a:=\rm const$, $a:=b$ (all types);\\                                      
+     $a:=a+b$, $a:=a*b$, $a:=a/b$ (integers and dimensions);\\                  
+     horizontal and vertical nesting of boxes                                   
+\item[3.] control constructs:\\                                                 
+     if-then-else statements testing relations between integers,                
+     dimensions, boxes, or boolean variables                                    
+\item[4.] modularization constructs:\\                                          
+     macros with up to 9~parameters (can be viewed as procedures without        
+     the concept of local variables).                                               
+\end{enumerate}                                                                 
+                                                                                
+Although the programming                                                      
+facilities of \TeX{} hardly exceed the abilities of a Turing machine,
+they are sufficient to
+handle relatively small programs. How about the graphics facilities?   
+Although \TeX{} has no built-in graphics facilities, it                         
+allows the placement of characters in arbitrary positions on                    
+the page. Therefore, complex pictures can be synthesized from elementary        
+picture elements treated as characters. Lamport has included such               
+a picture drawing environment in his macro package \LaTeX, using                
+quarter circles of different sizes and line segments (with and without          
+arrow heads) of different slopes as basic elements; see~\cite{LaTeX}.           
+These elements are sufficient for drawing trees.                                
+                                                                                
+This survey of \TeX's capabilities implies that \TeX{} may be a suitable        
+text processing system to implement a tree drawing algorithm directly.          
+We are basing our algorithm on the RT~algorithm, because this algorithm               
+gives the aesthetically most pleasing results. In the first version
+presented here, we         
+restrict ourselves to unary-binary trees, although our method is                
+applicable to arbitrary multiway trees. But in order to take advantage          
+of the text processing environment, we expand the algorithm to allow            
+labelled nodes.                                                                 
+                                                                                
+In contrast to previous tree drawing programs, we feel no necessity to          
+position the nodes of a tree on a fixed grid. While this may be                 
+reasonable for a plotter with a coarse resolution, it is certainly not          
+necessary for \TeX, a system that is capable of handling                        
+arbitrary dimensions                                                            
+and produces device \emph{independent} output.                            
+
+                                                                                
+\section{A representation method for \TeX{}trees}                               
+                                                                                
+The first problem to be solved in implementing our tree drawing algorithm       
+is how to choose a good internal representation                                 
+for trees. A straightforward adaptation                                         
+of the implementation by Reingold and Tilford requires, for each node,          
+at least the following fields:                                                  
+                                                                                
+\begin{enumerate}                                                               
+\item two pointers to the children of the node                                  
+\item two dimensions for the offset to the left and the right child (these      
+      may be different once there are labels of different widths to the         
+      left and right of the nodes)                                              
+\item two dimensions for the $x$- and $y$-coordinates of the final              
+      position of the nodes                                                     
+\item three or four labels                                                      
+\item one token to store the geometric shape (circle, square, framed text etc.)
+      of the node.                       
+\end{enumerate}                                                                 
+                                                                                
+Because these data are used very frequently in calculations, they should be     
+stored in registers (that's what variables are called in \TeX),
+rather than being recomputed, in order to obtain           
+reasonably fast performance. This gives a total of $10N$ registers for          
+a tree with $N$ nodes, which would exceed                                                          
+\TeX's limited supply of registers. Therefore, we present a
+modified algorithm hand-tailored to the abilities of \TeX{}.
+We start with the following observation.                   
+Suppose a unary-binary tree is constructed bottom-up, in a postorder            
+traversal. This is done by iterating the following three steps in               
+an order determined by the tree to be constructed.                              
+                                                                                
+\begin{enumerate}                                                               
+\item Create a new subtree consisting of one external node.                     
+\item Create a new subtree by appending the two subtrees created last           
+      to a new binary node; see \fig{Construct}.                               
+\item Create a new subtree by appending the subtree created last as a left,     
+      right, or unary subtree of a new node; see \fig{Construct}.              
+\end{enumerate}                                                                 
+                                                                                
+(A pointer to) each subtree that has been                                       
+created in steps 1--3 is pushed onto a stack, and                               
+steps 2 and 3 remove two trees or one, respectively,                            
+from the stack before the push                                                  
+operation is carried out. Finally, the tree to be constructed will              
+be the remaining tree on the 
+stack.                                             
+                                                                                
+\begin{Figure}                                                                  
+\centering                                                                      
+\begin{Tree}                                                                    
+\treesymbol{\lvls{2}}%                                                          
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}                    
+$+$                                                                             
+\treesymbol{\lvls{2}}%                                                          
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad               
+$\Longrightarrow$\quad                                                          
+\treesymbol{\lvls{2}}%                                                          
+\treesymbol{\lvls{2}}%                                                          
+\node{\type{dot}}%                                                              
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
+\end{Tree}
+
+\vskip\baselineskip                                                                      
+
+\begin{Tree}                                                                    
+\treesymbol{\lvls{2}}%                                                          
+\hspace{-\l@stlmoff}\usebox{\l@sttreebox}\hspace{\l@strmoff}\quad               
+$\Longrightarrow$\quad                                                          
+\treesymbol{\lvls{2}}%                                                          
+\node{\leftonly\type{dot}}%                                                     
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
+\quad or\quad                                                                   
+\treesymbol{\lvls{2}}%                                                          
+\node{\unary\type{dot}}%                                                        
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
+\quad or\quad                                                                   
+\treesymbol{\lvls{2}}%                                                          
+\node{\rightonly\type{dot}}%                                                    
+\hspace{-\l@stlmoff}\raisebox{\vd@st}{\usebox\l@sttreebox}\hspace{\l@strmoff}%  
+\end{Tree}                                                                      
+                                                                     
+\caption{Construction steps 2 and 3}                                                   
+\label{Construct}                                                              
+\end{Figure}                                                                    
+                                                                                
+This tree traversal is performed twice in the RT~algorithm.
+During the first pass,                            
+at each execution of step 2 or step 3, the relative positions of the         
+subtree(s) and of the new node are computed. 
+A closer examination of the RT~algorithm reveals that information about the
+subtree's coordinates is not needed during this pass; the contour information
+alone would be sufficient. Complete information is only needed in the second
+traversal, when the tree is actually drawn. Here a special feature of 
+\TeX{} comes in that allows us to save registers.
+Unlike Pascal, \TeX{} provides the capability of
+storing a drawing in a single box register that can be positioned freely in
+later drawings. This means that in our implementation the two passes
+of the original RT~algorithm can be intertwined into a single pass,
+storing for each subtree on the stack its contour and its drawing.
+Although the latter is a complex object, it takes only one of
+\TeX's precious registers.
+
+                                                                              
+\section{The internal representation}                                           
+                                                                                
+Given a tree, the corresponding \TeX{}tree is a box containing                  
+the ``drawing'' of the tree, together with some additional                      
+information about the contour of the tree.                                                              
+The reference point of a \TeX{}tree-box is always in the root of the            
+tree. The height, depth, and width of the box of a \TeX{}tree are               
+of no importance in this context.                                               
+                                                                                
+The additional information about the contour of the tree is stored in some      
+registers for numbers and dimensions and                                        
+is needed in order to put subtrees together to form a larger tree.              
+\var{loff} is an array of dimensions which contains for each                    
+level of the tree the horizontal offset between the                             
+left end of the                                                                 
+leftmost node at the current level and the                                      
+left end of the leftmost node at                                                
+the next level.                                                                 
+\var{lmoff} holds the horizontal offset between the root                        
+and the leftmost node of the whole tree. \var{lboff} holds the                  
+horizontal offset between the root and the leftmost node at                     
+the bottom level of the tree.                                                   
+Finally, \var{ltop} holds the distance between the reference point              
+of the tree and the leftmost end of the root.                                   
+The same is true for                                                            
+\var{roff}, \var{rmoff}, \var{rboff}, and \var{rtop}; just replace              
+``left'' by ``right''. Finally,                                                 
+\var{height} holds the height of the tree, and \var{type} holds the             
+geometric shape of the root of the tree. \fig{TeXtree} shows an example \TeX{}tree,
+i.e. a tree drawing and the corresponding additional information.
+
+\begin{Figure}
+\centering                                                                      
+\begin{Tree}                                                                    
+\e\ir\ir\e                                                                      
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
+  \i % A                                                                        
+\end{Tree}                                                                      
+\leavevmode                                                                     
+\stack{-10pt}{\vd@st}{%                                                
+     -10pt\\10pt\\10pt\\\var{loff}}%                              
+\hspace{1em}%                                                                   
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
+\hspace{1em}%                                                                   
+\stack{-10pt}{\vd@st}{%                                                         
+     15pt\\5pt\\-10pt\\\var{roff}}% 
+
+\vskip\baselineskip\raggedright
+height:~3, type:~dot, ltop:~2pt, rtop:~2pt, lmoff:~-10pt, rmoff:~20pt, lboff:~10pt,
+rboff:~10pt.
+
+\caption{A \TeX{}tree consists of the drawing of the tree and the
+additional information. The width of the dots is 4pt, the minimal separation between
+adjacent nodes is 16pt, making for a distance of 20pt center to center.
+The length of the small rule labelling one of the nodes is 5pt. The column left (right)
+of the tree drawing is the array \var{loff} (\var{roff}),
+describing the left (right) contour of the tree. At each level,
+the dimension given is the horizontal
+offset between the border at the current and at the next level. The offset between
+the left border of the root node and the leftmost node at level~1 is -10pt,
+the offset between the right border of the root node and the rightmost node at
+level~1 is 15pt, etc.}                               
+\label{TeXtree}
+\end{Figure}
+                                                                                
+Given two \TeX{}trees \var{A} and \var{B},                                      
+how can a new \TeX{}tree \var{C} be built that                                  
+consists of a new root and has \var{A} and \var{B} as subtrees?                 
+An example is given in \fig{AddInfo}.                        
+                                                                                
+\begin{Figure}                                                                  
+\centering                                                                      
+\begin{Tree}                                                                    
+\e\ir\ir\e                                                                      
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
+  \i % A                                                                        
+\end{Tree}                                                                      
+\leavevmode                                                                     
+A: \stack{-10pt}{\vd@st}{%                                                      
+     -10pt\\10pt\\10pt\\\ \\\var{loff}(\var{A})}%                              
+\hspace{1em}%                                                                   
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
+\hspace{1em}%                                                                   
+\stack{-10pt}{\vd@st}{%                                                         
+     15pt\\5pt\\-10pt\\\ \\\var{roff}(\var{A})}%                               
+\qquad                                                                          
+\begin{Tree}                                                                    
+\e\il\e\i\il\il\ir % B                                                          
+\end{Tree}                                                                      
+\leavevmode                                                                     
+B: \stack{-10pt}{\vd@st}{%                                                      
+     10pt\\-10pt\\-10pt\\-10pt\\-10pt\\\ \\\var{loff}(\var{B})}%               
+\hspace{1em}%                                                                   
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
+\hspace{1em}%                                                                   
+\stack{-10pt}{\vd@st}{%                                                         
+     10pt\\-10pt\\-10pt\\10pt\\-30pt\\\ \\\var{roff}(\var{B})}%                
+\\[\figspace]                                                                        
+\begin{Tree}                                                                    
+\e\ir\ir\e                                                                      
+  \node{\type{dot}\rightonly\rght{\unskip\vrule height.8pt width5pt depth0pt}}% 
+  \i % A                                                                        
+\e\il\e\i\il\il\ir % B                                                          
+\i % C                                                                          
+\end{Tree}                                                                      
+\leavevmode                                                                     
+C: \stack{-10pt}{\vd@st}{%                                                      
+     -20\\-10pt\\%                                                              
+     \makebox[0pt][r]{\var{loff}(\var{A})$\smash{\left\{\vrule height\vd@st     
+          depth\vd@st width0pt\right.}$ }%                                      
+     10pt\\10pt\\%                                                              
+     \makebox[0pt][r]{$\longrightarrow$ }%                                      
+     10pt\\%                                                                    
+     \makebox[0pt][r]{\raisebox{-.5\vd@st}{\var{loff}(\var{B})$\smash           
+          {\left\{\vrule height.5\vd@st                                         
+          depth.5\vd@st width0pt\right.}$ }}%                                   
+     \makebox[0pt][r]{-}10pt\\\ \\\var{loff}(\var{C})}%                        
+\hspace{1em}%                                                                   
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}%                         
+\hspace{1em}%                                                                   
+\stack{-10pt}{\vd@st}{%                                                         
+     20pt\\10pt\\-10pt\\-10pt%                                                  
+     \makebox[0pt][l]{\raisebox{-.5\vd@st}{                                     
+          $\smash{\left\}\vrule height2.5\vd@st                                 
+          depth2.5\vd@st width0pt\right.}$\var{roff}(\var{B})}}%                
+     \\10pt\\-30pt\\\ \\\var{roff}(\var{C})}%                                  
+                                                                                
+\vspace{\figspace}                                                              
+\centering                                                                      
+\begin{tabular}{|l|r|r|r|}                                                      
+\hline                                                                          
+&\multic{1}{c|}{\var{A}}&\multic{1}{c|}{\var{B}}&\multic{1}{c|}{\var{C}}\\      
+\hline                                                                          
+height&\multic{1}{c|}{3}&  \multic{1}{c|}{5}&  \multic{1}{c|}{6}\\              
+type&  \multic{1}{c|}{dot}&\multic{1}{c|}{dot}&\multic{1}{c|}{dot}\\            
+ltop&  2pt&                2pt&                2pt\\                            
+rtop&  2pt&                2pt&                2pt\\                            
+lmoff& -10pt&              -30pt&              -30pt\\                          
+rmoff& 20pt&               10pt&               30pt\\                           
+lboff& 10pt&               -30pt&              -10pt\\                          
+rboff& 10pt&               -30pt&              -10pt\\                          
+\hline                                                                          
+\end{tabular}\qquad                                                             
+\begin{tabular}{|c|r|r|}                                                        
+\hline                                                                          
+\multic{1}{|c|}{level}&\multic{1}{c|}{\var{totsep}}&                            
+                      \multic{1}{c|}{\var{currsep}}\\                           
+\hline                                                                          
+0&20pt&0/16pt\\                                                                 
+1&25pt&11/16\\                                                                
+2&40pt&1/16pt\\                                                                
+3&40pt&16pt\\                                                                
+\hline                                                                          
+\end{tabular}                                                                   
+\caption{The \TeX{}trees \var{A} and~\var{B} are combined to form the
+larger \TeX{}\-tree~\var{C}. The small table gives the 
+history of computation for \var{totsep} and \var{currsep}.}                  
+\label{AddInfo}                                                                 
+\end{Figure}                                                                    
+
+                                                                                
+First we determine which tree is higher; this is                                
+\var{B} in the example.                                                         
+Then we have to compute the minimal distance                                    
+between the roots of \var{A} and \var{B}, such that at all levels               
+of the trees there is free space of at least \var{minsep} between               
+the trees when they are drawn side by side.                                     
+For this purpose we keep track of two values, \var{totsep} and                  
+\var{currsep}. The variables \var{totsep} and \var{currsep}                     
+hold the total distance between the roots and the distance                      
+between the rightmost node of \var{A} and the leftmost node                     
+of \var{B} at the current level. In order to calculate                          
+\var{totsep} and \var{currsep}, we start at level 0 and                         
+visit each level of the trees until we reach the bottom level                   
+of the smaller tree; this is \var{A} in our example.                            
+                                                                                
+At level 0, the distance between the roots of \var{A} and \var{B}               
+should be at least \var{minsep}. Therefore, we set                              
+$\var{totsep}:=\var{minsep} + \var{rtop}(\var{A})                               
++ \var{ltop}(\var{B})$ and $\var{currsep}:=\var{minsep}$.                       
+Using $\var{roff}(\var{A})$ and $\var{loff}(\var{B})$, we can                   
+proceed to calculate \var{currsep} for the next level.                          
+If $\var{currsep} < \var{minsep}$, we have to increase \var{totsep} by          
+the difference and update \var{currsep}. This process is                        
+iterated until we reach the lowest level of \var{A}.                            
+Then \var{totsep} holds the final distance between the                          
+nodes of \var{A} and \var{B}, as calculated by the RT~algorithm. 
+If the root of \var{C} is a significant node, then the additional space ,
+which is 0pt by default, is added to \var{totsep}.               
+However, the approach of synthesizing                                           
+drawings from simple graphics characters allows only a finite                   
+number of orientations for the tree edges; therefore, \var{totsep}              
+must be increased slightly to fit the next orientation                          
+available.                                                                      
+                                                                                
+Now we are ready to construct the box of \TeX{}tree~\var{C}.                    
+Simply put \var{A} and~\var{B} side by side, with the reference                      
+points \var{totsep}~units apart, insert a new node                              
+above them, and connect the parent and children by edges.                       
+                                                                                
+Next, we update the additional information                                      
+for \var{C}. This can be done by using the additional information               
+for \var{A} and~\var{B}.                                                        
+Note that most components of $\var{roff}(\var{C})$ and                          
+$\var{lroff}(\var{C})$ are the same as in the higher tree, which                
+is \var{B} in our case.                                                         
+So, if we can avoid moving this information around, we only have                
+to access $\var{height}(\var{A}) + \var{const}$ many counters in                
+order to update the additional information for \var{C}.                         
+This implies that we can apply the same argument as                             
+in~\cite{TidierTrees}, which gives                                              
+us a running time of $\O(N)$ for drawing a tree with N nodes.                   
+                                                                                
+Therefore, we must carefully design the storage allocation for                  
+the additional information of \TeX{}trees in order to fulfill the               
+following requirements:                                                         
+If a new tree is built from                                                     
+two subtrees, the additional information of the new tree should                 
+share storage with its larger subtree.                                          
+Organizational overhead, that is,                                               
+pointers which keep track of the locations of different parts of additional     
+information, must be avoided.                                                   
+This means that all the additional information                                  
+for one \TeX{}tree should be stored in a row of consecutive dimension registers           
+such that only one pointer granting access to the first element                        
+in this row is needed.                                                          
+On the other hand, each parent                                                  
+tree is higher and therefore needs more storage than its subtrees.              
+So we must ensure that there is always enough space in the row                  
+for more information.                                                           
+                                                                                
+The obvious way to fulfill these requirements is to use a stack and to           
+allow only the topmost \TeX{}trees of this stack to be                          
+combined into a larger tree at any time.                   
+This leads to the following register allocation: A subsequent number of         
+box registers contains the treeboxes of the subtrees in the stack. A            
+subsequent number of token registers contains the type information for the      
+nodes of the subtrees in the stack. For each subtree in the stack,              
+a subsequent number of dimension registers contains the contour                 
+information of the subtree. The ordering of these groups of dimension           
+registers reflects the ordering of the subtrees in the                          
+stack. Finally, a subsequent number of counter registers contains               
+the height and the address of the first dimension register for                  
+each subtree in the stack. Four address counters store the addresses            
+of the last treebox, type information, height, and address of contour           
+information. A sketch of the register organization for a stack of \TeX{}trees
+is provided in \fig{Registers}.
+
+\begin{Figure}                                                                  
+Dimension registers\\                                                           
+\var{lmoff}(1) \var{rmoff}(1) \var{lboff}(1) \var{rboff}(1) \var{ltop}(1)       
+               \var{rtop}(1)\\                                                  
+\var{loff}($h_1$) \var{roff}($h_1$) \dots\ \var{loff}(1) \var{roff}(1)\\        
+\dots\\                                                                         
+\var{lmoff}($n$) \var{rmoff}($n$) \var{lboff}($n$) \var{rboff}($n$)             
+                 \var{ltop}($n$) \var{rtop}($n$)\\                              
+\var{loff}($h_n$) \var{roff}($h_n$) \dots\ \var{loff}(1) \var{roff}(1)\\        
+\ \\                                                                            
+Counter registers\\                                                             
+\var{lasttreebox} \var{lasttreeheight} \var{lasttreeinfo} \var{lasttreetype}\\  
+\var{treeheight}(1) \var{diminfo}(1) \dots\ \var{treeheight}($n$)               
+                    \var{diminfo}($n$)\\                                        
+\ \\                                                                            
+Box registers\\                                                                 
+\var{treebox}(1) \dots\ \var{treebox}($n$)\\                                    
+\ \\                                                                            
+Token registers\\                                                               
+\var{type}(1) \dots\ \var{type}($n$)
+
+\caption{\var{lasttreebox}, \var{lasttreeheight}, \var{lasttreeinfo},                    
+\var{lasttreetype} contain pointers to \var{treebox}($n$)                        
+\var{treeheight}($n$), \var{lmoff}($n$), \var{type}($n$),                       
+\var{diminfo}($i$) contains a pointer to                                        
+\var{lmoff}($i$). Unused dimension registers are                                
+allowed between the dimension registers of subsequent trees. The counter
+registers \var{lasttreebox},\ldots,\var{diminfo}($n$) serve as a directory
+mechanism to access the \TeX{}trees on the stack.}                   
+\label{Registers}                                                               
+\end{Figure}                                                                    
+                                                                   
+                                                                                
+When a new node is pushed onto the stack, the treebox, type information,        
+height, address of contour information, and contour information are             
+stored in the next free registers of the appropriate type, and the              
+four address counters are updated accordingly.                                  
+                                                                                
+When a new tree is formed from the topmost subtrees on the stack,               
+the treebox, type information, height, and address of contour information       
+of the new tree are sorted in the registers formerly used by the bottommost     
+subtree that has occured in the construction step, and the four address registers are       
+updated accordingly. This means that these informations for the subtrees        
+are no longer accessible. The contour information of the new subtree            
+is stored in the same registers as the contour information of the larger        
+subtree used in the construction, apart from the left and right offset          
+of the root to the left and right child, which are stored in the                
+following dimension registers. That means that gaps can occur                   
+between the contour information of subsequent subtrees in the                    
+stack, namely when the right subtree, which is on a higher position on the      
+stack, is higher than the left one. In order to avoid these                     
+gaps, the user can specify an option \verb.\lefttop. when entering a            
+binary node, which makes the topmost tree in the stack the                      
+left subtree of the node.                                                       
+                                                                                
+This stack concept also has consequences for the design of the user interface   
+that is discussed in Section~\ref{Interface}.                                   
+                                                                                
+\section{Space cost analysis}                                                   
+                                                                                
+Suppose we want to draw a unary-binary tree $T$ of height $h$ having            
+$N$ nodes\footnote{The height $h$ and the number of nodes $N$ refer to the
+drawing of the tree. $N$ is the number of circles, squares etc.~actually
+drawn, and $h$ is the number of levels in the drawing minus 1.}.
+According to our internal representation,                            
+for each subtree in the stack we need                                           
+                                                                                
+\begin{enumerate}                                                               
+\item one box register to store the box of the \TeX{}tree.                       
+\item one token register to store the type of the root of the subtree.           
+\item $2h^\prime+6$ dimension registers to store the additional 
+      information, where $h^\prime$ is the height of the                        
+      subtree.                                                                   
+\item three counter registers to store the register numbers of the             
+      box register, the token register, and the first dimension register above.         
+\end{enumerate}                                                                 
+                                                                                
+The following lemma relates to $h$ and $N$ the number
+of subtrees of $T$ which are on the      
+stack simultaneously and their heights.                          
+                                                                                
+\begin{lemma}                                                                   
+\begin{enumerate}                                                               
+\item At any time, there are at most $h+1$ subtrees of $T$ on the               
+      stack.                                                                    
+\item For each set $\T$ of subtrees of $T$ which are on the stack               
+      simultaneously we have                                                    
+      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)                             
+        \le\min(N,{(h+1)(h+2)\over2}).$$                                      
+\end{enumerate}                                                                 
+\end{lemma}                                                                     
+                                                                                
+\begin{proof}                                                                   
+\begin{enumerate}                                                               
+\item By induction on $h$.\label{stackdepth}                                           
+\item The trees in $\T$ are pairwise disjoint, and each tree of                 
+      height $h^\prime$ has at least $h^\prime+1$ nodes. This implies           
+      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)                             
+        \le N.$$                                                                
+      The second part is shown by induction on $h$.                           
+      The basis $h=0$ is clear.                                                 
+      Assume the assumption holds for all trees of height less than             
+      $h$. If $\T$                                                              
+      contains only subtrees of either the left or the right subtree            
+      of $T$, we have                                                           
+      $$\sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)\le                          
+      {h(h+1)\over2}\le{(h+1)(h+2)\over2}.$$                                    
+      Otherwise, $\T$ contains the left or the right subtree $T_s$ of           
+      $T$. Then all elements of $\T-\{T_s\}$ belong to the other                
+      subtree. This implies                                                     
+      \begin{eqnarray*}                                                         
+      \sum_{T^\prime\in \T}({\rm ht}(T^\prime)+1)&\le&                          
+      {\rm ht}(T_s)+1                                                           
+      +\sum_{T^\prime\in \T-\{T_s\}}({\rm ht}(T^\prime)+1)\\                    
+      &\le& h+{h(h+1)\over2}\le{(h+1)(h+2)\over2}.\proofend                       
+      \end{eqnarray*}                                                           
+\end{enumerate}                                                                 
+\end{proof}                                                                     
+                                                                                
+Therefore, our implementation uses at most $9h+2\min(N,(h+1)(h+2)/2)$          
+registers. In order to compare this with the                                    
+$10N$ registers used in the straightforward implementation,                     
+an estimation of the average height of a tree with $N$ nodes is                 
+needed. Several results, depending on the type of trees and of the                       
+randomization model, are cited in \fig{Stat}, which 
+compares the number of registers used in a straightforward           
+implementation with the average number of registers used in our                 
+implementation. This table shows clearly the advantage of our                  
+implementation. 
+
+\begin{Figure}
+\centering                                                                  
+\begin{tabular}{|c|c|c|c|c|}                                                    
+\hline                                                                          
+&registers&\multicolumn{3}{c|}{average registers}\\                             
+\cline{3-5}                                                                     
+nodes&(straight-&extended&unary-binary&binary\\                           
+&forward)&binary trees&trees&                                            
+     search trees\\
+&&($\sqrt{\pi n}$) \cite{AverageHeight}&
+  ($\sqrt{3\pi n}$) ~\cite{BinaryTrees}&
+  ($4.311\log n$) \cite{BinarySearchTrees}\\                                                             
+\hline                                                                          
+\ds8&    \ds80& \ds61.12& \ds94.15& \ds51.04\\                                  
+\ds9&    \ds90& \ds65.86&   100.89& \ds55.02\\                                  
+  10&      100& \ds70.44&   107.37& \ds58.80\\                                  
+  11&      110& \ds74.91&   113.64& \ds62.41\\                                  
+  12&      120& \ds79.26&   119.71& \ds65.87\\                                  
+  20&      200&   111.34&   163.56& \ds90.48\\                                  
+  30&      300&   147.37&   211.33&   117.31\\                                  
+  40&      400&   180.89&   254.75&   132.58\\                                  
+  50&      500&   212.80&   295.37&   143.54\\                                  
+\hline                                                                          
+\end{tabular}
+                                                                   
+\caption{The numbers of registers used by a straightforward implementation
+(second column) and by our modified implementation (third to fifth column)
+of the RT~algorithm are
+given for different types of trees and randomization models.
+The formula in parentheses indicates the average height of the respective class
+of trees, as depending on the number of nodes.}                                                        
+\label{Stat}                                                                    
+\end{Figure}                                                                    
+                                                                
+                                                                                
+\section{The user interface}\label{Interface}                                   
+                                                                                
+\subsection{General design considerations}                                      
+                                                                                
+The user interface of \TreeTeX{} has been designed in the spirit of             
+the thorough separation of the logical description of document components       
+and their layout; see~\cite{DocumentFormatting,GML}. This concept               
+ensures both uniformity and flexibility of document layout and frees            
+authors from layout problems which have nothing to do with the                  
+substance of their work. For some powerful implementations and projects         
+see \cite{Tables,Karlsruhe,LaTeX,Grif,Scribe}.                                  
+                                                                                
+In this context, the description of a tree is given in a purely                 
+logical form, and layout variations are defined by a separate style             
+command which is valid for all trees of a document.                             
+                                                                                
+A second design principle is to provide defaults for all specifications,        
+thereby allowing the user to omit many definitions                              
+if the defaults match what he or she wants.                                                        
+                                                                                
+The node descriptions of a tree must be entered in postorder.                   
+This fits the internal representation                                      
+of \TeX{}trees best. Although this is a natural method of describing a               
+tree, a user might prefer more flexible description methods.                     
+However, note that instances of well defined tree classes can be described      
+easily by \TeX{} macros. In section~\ref{ExampleClasses}. we give examples of macros                             
+for complete binary trees and Fibonacci trees.                                  
+                                                                                
+\TreeTeX{} uses the picture making macros of \LaTeX. If \TreeTeX{} is used with 
+any other macro package or format, the picture macros of                        
+\LaTeX{} are included automatically.                                            
+                                                                                
+\subsection{The description of a tree}                                          
+                                                                                
+The description of a tree is started by the command \verb.\beginTree.           
+and closed by \verb.\endTree. (or \verb.\begin{Tree}. and                       
+\verb.\end{Tree}. in \LaTeX). The description can be                  
+started in any mode; it defines a box and two dimensions. The                   
+box is stored in the box register \verb.\TeXTree. and contains the              
+drawing of the tree. The box has zero height and width, and its depth           
+is the height of the drawing. The reference point is in the                     
+center of the node of the tree. The dimensions are stored in the                
+registers \verb.\leftdist. and \verb.\rightdist. and describe                   
+the distance between the reference point and the left and                       
+right margin of the drawing. These data can be used to position the             
+drawing of the tree.                                                            
+                                                                                
+Note that the \TreeTeX{} macros don't contribute anything to the current        
+page but only store their results in the registers                              
+\verb.\TeXTree., \verb.\leftdist., and \verb.\rightdist.. It is the             
+user's job to put the drawing onto the page, using the                          
+commands \verb.\copy. or \verb.\box. (or \verb.\usebox. in \LaTeX).             
+                                                                                
+Each matching pair of \verb.\beginTree. and \verb.\endTree. must                
+contain the description for only \emph{one} tree.                               
+Descriptions of trees cannot be nested and                                      
+new registers cannot be allocated inside                                        
+a matching pair of \verb.\beginTree. and \verb.\endTree..                       
+                                                                                
+As already stated, each tree description defines the nodes of the tree in                        
+postorder, that is, a tree description is a particular sequence of node         
+descriptions.                                                                   
+                                                                                
+A node description, in turn, consists of the macro \verb.\node.,                
+followed by a list of node options, included in braces. The list                
+of node options may be empty. The node options describe the labels,             
+the geometric shape (type), and the outdegree of the node. Default values are   
+provided for all options which are not explicitly specified.                    
+The following node options are available:                                       
+                                                                                
+\begin{enumerate}                                                               
+\item[1.] \verb.\lft{<label>}., \verb.\rght{<label>}.,                          
+     \verb.\cntr{<label>}.,\\\verb.\bnth{<label>}.:\\                           
+     These options describe the labels which are put to the left of, to the     
+     right of, in the                                                           
+     center of, or beneath the node (the latter only makes sense for            
+     external nodes). The arguments of these macros are processed in            
+     internal horizontal mode (LR-mode in \LaTeX), but can consist of           
+     arbitrary nested boxes for more sophisticated labels. For each of          
+     these options, the default is an empty label.                              
+                                                                                
+\item[2.] \verb.\external., \verb.\unary.,                                      
+     \verb.\leftonly., \verb.\rightonly.:\\                                     
+     These options describe the outdegree                                       
+     of the node.                                   
+     The default is binary (no outdegree option is specified).                  
+                                                                                
+\item[3.] \verb.\type{<type>}.:\\                                               
+     This option describes the type or geometric shape of the node.             
+     \verb.<type>. can have the values \verb.square.,                         
+     \verb.dot., \verb.text., or \verb.frame.. 
+     The default value is \verb.circle. (no type is specified). A node of type
+     \verb.square. has a fixed width, while a node of type \verb.frame. has its
+     width determined by the center label. A node of type \verb.text. has no frame
+     around its center label. The center label can have arbitrary width.
+
+\item[3.] \verb.\leftthick., \verb.\rightthick.:
+     These options change the thickness of the left or right outgoing edge of
+     a binary node. Defaults are thin edges (neither option is specified).      
+                                                                                
+\item[4.] \verb.\lefttop.:\\                                                    
+     The node option \verb.\lefttop. in a binary node makes the                 
+     last entered subtree the left child of the node (the right child is the    
+     default). This option helps to cut down on the number of dimension registers  
+     used during the construction of a tree. As a rule of thumb,                
+     this option is recommended when the left subtree has a smaller             
+     height than the right subtree, that is,                                    
+     in this case the right subtree should                                      
+     be entered before the left one and their parent should be assigned the option 
+     \verb.\lefttop..                                                           
+\end{enumerate}                                                                 
+                                                                                
+\subsection{Macros for classes of trees}
+\label{ExampleClasses}                                        
+                                                                                
+Tree descriptions can be produced by macros. This is especially useful          
+for trees which belong to a larger class of trees and which can be specified    
+by some simple parameters. A small library of such                              
+macros is provided in the file \verb!TreeClasses.tex!.                          
+                                                                                
+\begin{enumerate}                                                               
+\item[1.] \verb.\treesymbol{<node options>}.:\\                                 
+     This macro produces a triangular tree symbol which can be included in      
+     a tree description instead of an external node. Labels for these           
+     tree symbols are described as for ordinary nodes. In addition, the         
+     options \verb.\lvls{<number>}. and \verb.\slnt{<number>}.                  
+     are provided. \verb.\lvls. defines the number of levels in the             
+     tree over which the triangle extends, and \verb.\slnt. gives               
+     the slant of the sides of the triangle, ranging from 1~(minimal)           
+     to 24~(maximal). On the other hand,                                        
+     \verb.\treesymbol. does not expand to a tree description, because          
+     a tree symbol cannot be built from subtrees, and, on the other hand,       
+     it is not a node, because it is allowed to extend over several tree        
+     levels and therefore has a longer contour than an ordinary node.           
+                                                                                
+\item[2.] \verb.\binary{<bin specification>}.:\\                                
+     This macro truly expands to a tree description. It produces                
+     a complete binary tree, that is, an extended binary tree,                  
+     where, for a given $h$, all external nodes appear at level $h$             
+     or $h-1$, and all external nodes at level $h$ lie left of those at         
+     level $h-1$. \verb.<bin specification>. consists of the                    
+     following options:                                                         
+     \verb.\no{<number>}. defines the number of internal nodes,                 
+     with \verb.<number>. greater than 0, and                                   
+     \verb.\squareleaves. produces leaves of type                               
+     \verb.square.. Defaults are \verb.\no{1}. and                              
+     leaves of type \verb.circle..                                              
+                                                                                
+\item[3.] \verb.fibonacci{<fib specification>}.:\\                              
+     This macro produces a Fibonacci tree.              
+     \verb.<fib specification>. allows for the three options                    
+     \verb.\hght{<number>}., \verb.\unarynodes.,                                 
+     and \verb.\squareleaves..                                                  
+     Normally, a Fibonacci tree of height $h+2$ is a binary tree                
+     with Fibonacci trees of height $h$ and $h+1$ as left and                   
+     right subtrees. The option \verb.\unarynodes. means that the               
+     Fibonacci tree is augmented by unary nodes such that each                  
+     two subtree siblings have the same height. These are examples
+     of what has been called brother-trees in the literature; 
+     see~\cite{Brother}. Defaults are                    
+     \verb.\hght{0}., the unaugmented version of a Fibonacci tree,              
+     and external nodes of type \verb.circle..                                  
+\end{enumerate}                                                                 
+                                                                                
+\subsection{Style options for trees}
+\sloppy                                            
+The \TreeTeX{} package includes a style command                           
+\verb.\Treestyle{<style option>}., where \verb.<style option>.                   
+contains all the parameter settings the user might want                         
+to change.                                                                      
+Normally, the command \verb.\Treestyle. appears only once at the beginning      
+of the document and the style options are valid for all trees of the            
+document.      
+
+\fussy                                                                                
+The changes in the style options are global. A \verb.\Treestyle. command        
+changes only the specified style options; non-specified options retain          
+the last specified value or the default value, respectively. The following            
+style options are available:                                                    
+                                                                                
+\begin{enumerate}                                                               
+\item[1.] \verb.\treefonts{<font options>}.:\\                                  
+     \sloppy                                                                    
+     \verb.\treefonts. is invoked by \verb.\beginTree., and it simply executes  
+     whatever is specified in \verb.<font options>.. Defaults are               
+     \verb.\treefonts{\tenrm}. (or \verb.\treefonts{\normalsize\rm}. in         
+     \LaTeX).                                                                   
+                                                                                
+\fussy                                                                          
+\item[2.] \verb.\nodesize{<size>}.:\\                                           
+     \verb.\nodesize. defines the size of the nodes. \verb.<size>. is a         
+     dimension and specifies the diameter of circle nodes. The                  
+     width of square nodes is adjusted accordingly to be slightly               
+     smaller than the diameter of circle nodes in order to                      
+     balance their appearance. Furthermore,                                     
+     \verb.\nodesize. adjusts the amount of space by which the                  
+     baseline of the labels is placed beneath the center of the node.           
+     The default value of \verb.\nodesize. suits the default of                 
+     \verb.\treefonts. (taking into account the size option                     
+     of \LaTeX's document style).                                               
+                                                                                
+\item[3.] \verb.\vdist{<dimen>}., \verb.\minsep{<dimen>}.,                      
+     \verb.\addsep{<dimen>}.:\\                                                 
+     \sloppy                                                                    
+     \verb.vdist. specifies the vertical distance between two                   
+     subsequent levels of the tree. Default is \verb.\vdist{60pt}..             
+     \verb.\minsep. specifies the minimal horizontal distance between two       
+     adjacent nodes. Default is \verb.\minsep{20pt}..                           
+     \verb.\addsep. specifies the additional amount of horizontal space         
+     by which two subtree siblings are pushed apart farther than                
+     calculated by the RT~algorithm,                                            
+     if the level at which they are closest is beneath                          
+     their root level. Default is \verb.\addsep{0pt}.                           
+                                                                                
+\fussy                                                                          
+\item[4.] \verb.\extended., \verb.\nonextended.:\\                              
+     With the option \verb.\extended. in effect, the nodes of a binary          
+     tree are placed in exactly the same way as they would be in the            
+     associated extended version of the tree (the missing nodes are             
+     assumed to have no labels). The default is \verb.\nonextended.,            
+     that is the usual layout.                                                  
+\end{enumerate}                                                                 
+                                                                                
+Some examples of tree descriptions                                              
+are given in the next figures.                
+A detailed description of the                                                   
+\TreeTeX{} macros is given in~\cite{TreeTeX}.
+                                   
+\Treestyle{\vdist{60pt}}                                                        
+\dummyhalfcenterdim@n=10pt                                                      
+                                                                                
+\begin{Figure}
+\centering
+\begin{Tree}
+\node{\external\bnth{first}\cntr{1}\lft{Beeton}}                         
+\node{\external\cntr{3}\rght{Kellermann}}                                       
+\node{\cntr{2}\lft{Carnes}}                                                     
+\node{\external\cntr{6}\lft{Plass}}                                             
+\node{\external\bnth{last}\cntr{8}\rght{Tobin}}                          
+\node{\cntr{7}\rght{Spivak}}                                                    
+\node{\leftonly\cntr{5}\rght{Lamport}}                                          
+\node{\cntr{4}\rght{Knuth}}                                                     
+\end{Tree}
+                                                                        
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
+
+\begin{verbatim}
+\begin{Tree}
+\node{\external\bnth{first}\cntr{1}\lft{Beeton}}                         
+\node{\external\cntr{3}\rght{Kellermann}}                                       
+\node{\cntr{2}\lft{Carnes}}                                                     
+\node{\external\cntr{6}\lft{Plass}}                                             
+\node{\external\bnth{last}\cntr{8}\rght{Tobin}}                          
+\node{\cntr{7}\rght{Spivak}}                                                    
+\node{\leftonly\cntr{5}\rght{Lamport}}                                          
+\node{\cntr{4}\rght{Knuth}}                                                     
+\end{Tree}
+                                                                        
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim} 
+
+\caption{This is an example of a tree that includes labels.}                                         
+\end{Figure}
+
+\begin{Figure}
+\centering
+\begin{Tree}
+\node{\external\type{frame}\bnth{first}\cntr{Beeton}}                         
+\node{\external\type{frame}\cntr{Kellermann}}                                       
+\node{\type{frame}\cntr{Carnes}}                                                     
+\node{\external\type{frame}\cntr{Plass}}                                             
+\node{\external\type{frame}\bnth{last}\cntr{Tobin}}                          
+\node{\type{frame}\cntr{Spivak}}                                                    
+\node{\leftonly\type{frame}\cntr{Lamport}}                                          
+\node{\type{frame}\cntr{Knuth}}                                                     
+\end{Tree}
+                                                                        
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
+
+\begin{verbatim}
+\begin{Tree}
+\node{\external\type{frame}\bnth{first}\cntr{Beeton}}                         
+\node{\external\type{frame}\cntr{Kellermann}}                                       
+\node{\type{frame}\cntr{Carnes}}                                                     
+\node{\external\type{frame}\cntr{Plass}}                                             
+\node{\external\type{frame}\bnth{last}\cntr{Tobin}}                          
+\node{\type{frame}\cntr{Spivak}}                                                    
+\node{\leftonly\type{frame}\cntr{Lamport}}                                          
+\node{\type{frame}\cntr{Knuth}}                                                     
+\end{Tree}
+                                                                        
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim}
+\caption{This is an example of a tree with framed center labels.} 
+\end{Figure}
+
+\begin{Figure}                                                                  
+\centering                                                                      
+\begin{Tree}                                                                    
+\binary{\no{6}\squareleaves}                                                    
+\end{Tree}
+                                                                      
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
+                                                                     
+\begin{verbatim}                                                                
+\begin{Tree}                                                                    
+\binary{\no{6}\squareleaves}                                                    
+\end{Tree}
+                                                                      
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim}                                                                  
+\caption{This is an example of a complete binary tree.}                                     
+\end{Figure} 
+
+                                                                   
+\begin{Figure}                                                                  
+\centering                                                                      
+\begin{Tree}                                                                    
+\fibonacci{\hght{4}\unarynodes\squareleaves}                                    
+\end{Tree}
+
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}\                         
+                                                                      
+\begin{verbatim}                                                                
+\begin{Tree}                                                                    
+\fibonacci{\hght{4}\unarynodes\squareleaves}                                    
+\end{Tree}
+\hspace{\leftdist}\usebox{\TeXTree}\hspace{\rightdist}
+\end{verbatim}
+                                                                  
+\caption{This is an example of a Fibonacci tree.}                                           
+\end{Figure}                                                                    
+ 
+                                                                                
+\clearpage                                                                      
+\bibliography{trees}                                                            
+\end{document}                                                                  
+                                                                                
+                                                                                
+