%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% mptrees.mp %% %% Probability trees with MetaPost %% %% o.peault@posteo.net %% %% Version 21.11 (November 2021) %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% string mptreesversion; mptreesversion:="21.11"; % % This work may be distributed and/or modified under the conditions of % the LaTeX Project Public License, either version 1.3 of this license % or (at your option) any later version. The latest version of this % license is in http://www.latex-project.org/lppl.txt % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% message "mptrees version " & mptreesversion; if not known mplib: input latexmp fi; boolean arbres_latexmp; arbres_latexmp:=true; numeric posprob; % position de l'étiquette sur la branche posprob:=0.6; numeric scaleprob; scaleprob:=0.85; % échelle de l'étiquette sur la branche numeric scaleev; scaleev:=1; % échelle de l'évènement numeric dirtree,dirlabel; % direction de l'arbre dirtree:=0; dirlabel:=0; numeric proboffset; % redéfinition temporaire de labeloffset pour affichage probas proboffset:=labeloffset; numeric endedgeshift; endedgeshift:=0; numeric brokenlineratio; brokenlineratio:=0.2; pair Orig_arbre[][]; % sommet de l'arbre i,j Orig_arbre[1][1]:=origin; numeric colonne_cour,ligne_cour; numeric countligne[]; % nombre de ligne pour le niveau i numeric typedec; % 0 origine dépend de la taille de l'évènement, 1 taille fixée numeric shiftev; % taille du décalage dans le cas où typedec=1 shiftev:=0; % dans ce cas, typedec =0 extra_beginfig:=extra_beginfig & "shiftev:=0;"; % remise à zéro en début de figure numeric typeprob; % affichage de la proba sur la branche typeprob:=1; boolean abscoord; % Coordonnée absolues abscoord=false; string branchtype,endlabeltype; branchtype="segment"; endlabeltype="none"; boolean edgearrow; edgearrow:=false; numeric endlabelspace; endlabelspace:=1cm; % Étiquette au début de l'arbre picture mptreestartlabel; mptreestartlabel:=nullpicture; vardef startlabel(expr s)= mptreestartlabel:=labelarbres(s); if dirtree<>0: dirlabel:=dirtree fi; if dirlabel>0: theevlabel(labelarbres(s) scaled scaleev,origin,-180+dirlabel) else: theevlabel(labelarbres(s) scaled scaleev,origin,180+dirlabel) fi enddef; % Étiquettes de fin d'arbre vardef colonne.endlabel[]= @ enddef; vardef endlabel[][](expr s)= save i,j,tmpbranchtype,tmpabscoord,tmps; numeric i,j; string tmpbranchtype,tmps; boolean tmpabscoord; i:=colonne.#@; j:=@; if string s: tmps:= "\strut \phantom{.}" & s else: picture tmps;tmps:=s fi; tmpbranchtype:=branchtype; branchtype:=endlabeltype; tmpabscoord:=abscoord; abscoord:=false; tree[i][j]((endlabelspace,0) rotated (dirlabel-dirtree))(tmps," ") hide(branchtype:=tmpbranchtype;abscoord:=tmpabscoord;) enddef; vardef labelarbres(expr s)= save p; picture p; if picture s: p=s elseif path s: p=image(draw s) else: p=textext(s) fi; p enddef; % longueur du segment qui traverse la figure p selon l'angle inc vardef longdir(expr p,inc)= if urcorner p = llcorner p: 0 elseif (abs inc <= abs angle (urcorner p -llcorner p)) or (abs inc > 180 - abs angle (urcorner p -llcorner p)): abs((lrcorner p - llcorner p)/cosd(inc)) else: abs((ulcorner p - llcorner p)/sind(inc)) fi enddef; vardef theevlabel(expr s,z,inc)= save p; picture p; save d; numeric d; if picture s: p=s else: p = textext(s) fi; d=0.5*longdir(p,inc); p shifted (z - center p + (d+labeloffset)*dir(inc)) enddef; def evlabel = draw theevlabel enddef; vardef thelabelbranchehaut(expr Ori,Fin,Fig)= interim labeloffset:=proboffset; save M,pp,tt; pair M; path pp;numeric tt; pp=dessinbranche(Ori,Fin); tt = arctime posprob*(arclength pp) of pp; M= point tt of pp; M:=M + 0.5(llcorner Fig - lrcorner Fig); if (angle(direction tt of pp) <= 90) and (angle(direction tt of pp) > -90): M:=M+proboffset*unitvector((direction tt of pp) rotated 90) else: M:=M-proboffset*unitvector((direction tt of pp) rotated 90) fi; Fig shifted M enddef; vardef thelabelbranchehautrot(expr Ori,Fin,Fig)= interim labeloffset:=proboffset; save M,pp,tt; pair M; path pp;numeric tt; pp=dessinbranche(Ori,Fin); tt = arctime posprob*(arclength pp) of pp; M= point tt of pp; if xpart Ori <= xpart Fin: Fig shifted ( - 0.5*lrcorner Fig- 0.5* llcorner Fig+(0,proboffset)) rotated angle(direction tt of pp) shifted M else: Fig shifted ( - 0.5*lrcorner Fig- 0.5* llcorner Fig+(0,proboffset)) rotated (-180+angle(direction tt of pp)) shifted M fi enddef; vardef thelabelbranchesuper(expr Ori,Fin,Fig)= save figtmp,M,pp; pair M; path pp; picture figtmp; pp=dessinbranche(Ori,Fin); M= point (arctime posprob*(arclength pp) of pp) of pp; figtmp:=Fig shifted (M-0.5*(lrcorner Fig + ulcorner Fig)); image(fill bbox figtmp withcolor background; draw figtmp) enddef; vardef thelabelbranchesuperrot(expr Ori,Fin,Fig)= save figtmp; picture figtmp; save M,pp,tt; pair M; path pp;numeric tt; pp=dessinbranche(Ori,Fin); tt = arctime posprob*(arclength pp) of pp; M= point tt of pp; figtmp:=Fig shifted (-0.5*(lrcorner Fig + ulcorner Fig)) rotated angle(direction tt of pp) shifted M; image(fill bbox figtmp withcolor background; draw figtmp) enddef; vardef thelabelbranche(expr Ori,Fin,Fig)= if typeprob=1: thelabelbranchehaut(Ori,Fin,Fig) elseif typeprob=2: thelabelbranchesuper(Ori,Fin,Fig) elseif typeprob=3: thelabelbranchehautrot(Ori,Fin,Fig) elseif typeprob=4: thelabelbranchesuperrot(Ori,Fin,Fig) fi enddef; vardef brokenline(expr Ori,Fin)= save B,M; pair B,M; B=(xpart (Fin rotated -dirlabel), ypart (Ori rotated -dirlabel)); M = (brokenlineratio*abs(B-(Ori rotated -dirlabel)),0) rotated dirlabel; Ori -- (Ori + M) -- (Fin - M) -- Fin enddef; %%%%%%%% Dessin de la branche vardef dessinbranche(expr Ori,Fin)= if branchtype="segment": Ori--Fin elseif branchtype="curve": Ori{dir dirlabel}..{dir dirlabel}Fin elseif branchtype="broken": brokenline(Ori,Fin) else: Ori--Fin fi enddef; %%%%%%%% Dessin complet de la branche (avec proba, évènement vardef branche_abs(expr Ori,Fin,Eve,Pro)= save figtmp,thelab,figlab,tmpshft; picture figtmp,thelab,figlab; numeric tmpshft; thelab:=labelarbres(Pro) scaled scaleprob; if ypart Ori > ypart Fin: %le décalage se fait en fonction de la direction de la branche tmpshft:=endedgeshift else: tmpshft:=-endedgeshift fi; figlab:=thelabelbranche(Ori,(Fin+ (0,tmpshft)) rotatedaround(Ori,dirtree),thelab); figtmp=image(% if branchtype<>"none": if edgearrow: drawarrow else: draw fi % dessinbranche(Ori,Fin + (0,tmpshft)) rotatedaround(Ori,dirtree) dessinbranche(Ori,(Fin + (0,tmpshft)) rotatedaround(Ori,dirtree)) fi; draw figlab; evlabel(Eve,Fin rotatedaround(Ori,dirtree),dirlabel) ); figtmp enddef; vardef branche_rel(expr Ori,dx,dy,Eve,Pro)= branche_abs(Ori,Ori+(dx,dy),Eve,Pro) enddef; vardef _arbre_(expr Ori)(text t)(text r)= % (pt)(pt1,pt2...)(ev,proba,ev,proba...) save figtmp,n,Fintmp,labeltmp,counttmp,countlab; picture figtmp,labeltmp[]; numeric counttmp,countlab; pair Fintmp[]; if dirtree<>0: dirlabel:=dirtree fi; if colonne_cour=1: for i=1 upto 128: countligne[i]:=0; endfor fi figtmp=image(% counttmp:=0; for i=t: counttmp:=counttmp+1; Fintmp[counttmp]:=i; endfor; counttmp:=0;countlab:=0; for i=r: countlab:=countlab+1; labeltmp[countlab]:=labelarbres(i); if countlab=2: counttmp:=counttmp+1;countligne[colonne_cour+1]:=countligne[colonne_cour+1]+1; draw branche_abs(Ori,Fintmp[counttmp],labeltmp[1] scaled scaleev,labeltmp[2]); Orig_arbre[colonne_cour+1][countligne[colonne_cour+1]]:=Fintmp[counttmp] shifted if typedec=0: ((longdir(labeltmp[1] scaled scaleev,dirlabel)+2labeloffset)*(1,0)) else: (shiftev,0) fi if dirtree=0: rotatedaround(Fintmp[counttmp],dirlabel) fi rotatedaround(Ori,dirtree); countlab:=0 fi; endfor); figtmp enddef; vardef arbre_var(expr Ori,horiz,Ymax)(text t)(text r)= % (pt,dec horiz,Ymax)(dy1,dy2...)(ev,proba,ev,proba...) save Fin,compt,Ytmp; numeric compt,Ytmp; pair Fin[]; Fin[0]:=Ori+(horiz,Ymax); compt:=0;Ytmp:=Ymax; for i=t: compt:=compt+1;Ytmp:=Ytmp-i; Fin[compt]:=Ori+(horiz,Ytmp); endfor def ptarr= Fin[0] for i=1 upto compt: ,Fin[i] endfor enddef; _arbre_(Ori)(ptarr)(r) enddef; vardef arbre_decv(expr Ori,horiz,Ymax)(text t)(text r)= % (pt,dec horiz)(dy1,dy2...)(ev,proba,ev,proba...) save Fin,compt,Ytmp; numeric compt,Ytmp; pair Fin[]; if abscoord: Fin[0]:=(horiz,Ymax) else: Fin[0]:=Ori+(horiz,Ymax) fi; compt:=0;Ytmp:=Ymax; for i=t: compt:=compt+1;Ytmp:=i; if abscoord: Fin[compt]:=(horiz,Ytmp) else: Fin[compt]:=Ori+(horiz,Ytmp) fi; endfor def ptarr= Fin[0] for i=1 upto compt: ,Fin[i] endfor enddef; _arbre_(Ori)(ptarr)(r) enddef; vardef arbre_fixe(expr Ori,horiz,Ymax,decy)(text r)= save Fin,compt,compt_par; numeric compt,compt_par; pair Fin[]; if abscoord: Fin[0]:=(horiz,Ymax + ypart Ori) else: Fin[0]:=Ori+(horiz,Ymax) fi; compt:=0;compt_par:=0; for i=r: compt:=compt+1;compt_par:=compt_par+1; if compt_par=2: compt_par:=0; if abscoord: Fin[compt/2]:=(0, ypart Ori) + (horiz,Ymax-decy*compt/2) else: Fin[compt/2]:=Ori+(horiz,Ymax-decy*compt/2) fi; fi endfor def ptarr= Fin[0] for i=1 upto compt: ,Fin[i] endfor enddef; _arbre_(Ori)(ptarr)(r) enddef; vardef colonne.reg.arbre[]= @ enddef; vardef colonne.comp.arbre[]= @ enddef; vardef colonne.dec.arbre[]= @ enddef; vardef reg.arbre[][](text t)(text r)= save compt,decal; numeric compt,decal[]; colonne_cour:=colonne.#@; ligne_cour:=@; typedec:=0; if shiftev<>0: typedec:=1 fi; compt:=0; for i=t: if numeric i: compt:=compt+1; decal[compt]:=i else: typedec:=1; scantokens i fi; endfor compt:=0; for i=r: compt:=compt+1; endfor arbre_fixe(Orig_arbre[colonne_cour][ligne_cour],decal[1],(compt/2-1)*decal[2]/2,decal[2])(r) enddef; vardef comp.arbre[][](text t)(text r)= colonne_cour:=colonne.#@; ligne_cour:=@; typedec:=0; if shiftev<>0: typedec:=1 fi; save tmp_compt; numeric tmp_compt; tmp_compt:=0; for i=t: if string i: typedec:=1; scantokens i fi; endfor if abscoord: def tmp_text=t enddef else: def tmp_text= % Si abscoord tmp_text=t for i=t: hide(tmp_compt:=tmp_compt+1;) if (tmp_compt>1) and (not string i): ,Orig_arbre[colonne_cour][ligne_cour]+i elseif not string i: Orig_arbre[colonne_cour][ligne_cour]+i fi endfor enddef fi; _arbre_(Orig_arbre[colonne_cour][ligne_cour])(tmp_text)(r) enddef; %%%%%%%%%%%%%%%%%%%%%%%%% Modif 10/16 - Décalage par rapport à l'origine vardef dec.arbre[][](text t)(text r)= save decal; numeric decal[]; colonne_cour:=colonne.#@; ligne_cour:=@; typedec:=0; if shiftev<>0: typedec:=1 fi; save tmp_compt; numeric tmp_compt; tmp_compt:=0; for i=t: if numeric i: tmp_compt:=tmp_compt+1; decal[tmp_compt]:=i else: typedec:=1; scantokens i fi; endfor def tmp_text= decal[3] for i=4 upto tmp_compt: ,decal[i] endfor enddef; arbre_decv(Orig_arbre[colonne_cour][ligne_cour],decal[1],decal[2])(tmp_text)(r) enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 vardef arbre[][](text t)(text r)= save tmp_type,tmp_compt; numeric tmp_type,tmp_compt; tmp_compt:=0; for i=t: if numeric i: tmp_compt:=tmp_compt+1 fi; endfor if tmp_compt=0: % Arbre défini par des points (dec_horiz,dec_vert) comp.#@.@(t)(r) elseif tmp_compt=2: % Arbre régulier, on donne largeur et esp. vert. reg.#@.@(t)(r) else: dec.#@.@(t)(r) % Arbre semi-régulier, largeur fixe, decalages vert. fi enddef; def tree = arbre enddef; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%% Binomial string bernoullisuccessevent, bernoullifailureevent, bernoullisuccessprob, bernoullifailureprob; bernoullisuccessevent := "$S$"; bernoullifailureevent := "$\overline{S}$"; bernoullisuccessprob := "$p$"; bernoullifailureprob := "$q$"; numeric bernoulliscalebranch; bernoulliscalebranch := 0.8; vardef bernoulliprocess(expr n)(expr l,h)= save tmp,ll,shiftev; numeric ll; shiftev:=max(longdir(labelarbres(bernoullisuccessevent),dirtree),longdir(labelarbres(bernoullifailureevent),dirtree))+2*labeloffset; ll := l; picture tmp; tmp := image(% for i := 1 upto n: for j := 1 upto (2**(i-1)): draw tree[i][j](ll,h*(2**(n-i)))(bernoullisuccessevent, bernoullisuccessprob, bernoullifailureevent, bernoullifailureprob); endfor ll := ll * bernoulliscalebranch; endfor ); tmp enddef; vardef bernoulliprocessL(expr n)(expr L,H)= save l,h,lpic,hpic; numeric l,h,lpic,hpic; hpic:=0.5*longdir(labelarbres(bernoullisuccessevent),90+dirtree) + 0.5*longdir(labelarbres(bernoullifailureevent),90+dirtree); h := (H-hpic)/((2**n)-1); lpic:=longdir(mptreestartlabel,dirtree) + n*max(longdir(labelarbres(bernoullisuccessevent),dirtree) , longdir(labelarbres(bernoullifailureevent),dirtree)) + (2*n-1)*labeloffset; if bernoulliscalebranch<>1: l:= (L-lpic)*(1-bernoulliscalebranch)/(1-bernoulliscalebranch**n) else: l:= (L-lpic)/n fi; bernoulliprocess(n)(l,h) enddef; vardef binomialtree(expr n)(expr l,h)= save tmp,ll,posprob,shiftev,bernoulliscalebranch; numeric ll; posprob:=0.5;bernoulliscalebranch:=1; shiftev:=longdir(labelarbres("3"),dirtree)+2*labeloffset; ll := l; picture tmp; tmp := image(% draw tree[1][1](ll,h)("$1$", bernoullisuccessprob, "$0$", bernoullifailureprob); if n>1: for i := 2 upto n: for j := 1 upto (i-1): draw tree[i][2*j-1](ll,h)(decimal(i-j+1), bernoullisuccessprob, " ", bernoullifailureprob); endfor draw tree[i][2*i-2](ll,h)("$1$", bernoullisuccessprob, "$0$", bernoullifailureprob); endfor ll := ll * bernoulliscalebranch; fi ); tmp enddef; vardef binomialtreeL(expr n)(expr L,H)= save l,h,lpic,hpic; numeric l,h,lpic,hpic; hpic:=longdir(labelarbres("0"),90+dirtree); h := (H-hpic)/n; lpic:=longdir(mptreestartlabel,dirtree) + n*max(longdir(labelarbres("0"),dirtree) , longdir(labelarbres("0"),dirtree)) + (2*n-1)*labeloffset; l:= (L-lpic)/n; binomialtree(n)(l,h) enddef;