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position","\u002Fsnt-2nde\u002Fse-localisation-cartographie\u002Fconfidentialite-position","0.snt-2nde\u002Fse-localisation-cartographie\u002F5.confidentialite-position",{"title":145,"path":146,"stem":147,"children":148},"Informatique embarquée et objets connectés","\u002Fsnt-2nde\u002Fsf-info-embarquee","0.snt-2nde\u002Fsf-info-embarquee\u002F0.index",[149,150,154,158],{"title":145,"path":146,"stem":147},{"title":151,"path":152,"stem":153},"Systèmes informatiques embarqués","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fsystemes-embarques","0.snt-2nde\u002Fsf-info-embarquee\u002F1.systemes-embarques",{"title":155,"path":156,"stem":157},"Capteurs et actionneurs","\u002Fsnt-2nde\u002Fsf-info-embarquee\u002Fcapteurs-actionneurs","0.snt-2nde\u002Fsf-info-embarquee\u002F2.capteurs-actionneurs",{"title":159,"path":160,"stem":161},"IHM et objets 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Python","\u002Fsnt-2nde\u002Fsg-photo-numerique\u002Ftraitement-rgb","0.snt-2nde\u002Fsg-photo-numerique\u002F4.traitement-rgb",{"title":185,"path":186,"stem":187,"children":188},"Notions transversales de programmation","\u002Fsnt-2nde\u002Fsh-prog-notions","0.snt-2nde\u002Fsh-prog-notions\u002F0.index",[189,190,194,198,202],{"title":185,"path":186,"stem":187},{"title":191,"path":192,"stem":193},"Séquences d'instructions et variables","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fsequences-affectations","0.snt-2nde\u002Fsh-prog-notions\u002F1.sequences-affectations",{"title":195,"path":196,"stem":197},"Instructions conditionnelles","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fconditionnelles","0.snt-2nde\u002Fsh-prog-notions\u002F2.conditionnelles",{"title":199,"path":200,"stem":201},"Boucles bornées et non bornées","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Fboucles","0.snt-2nde\u002Fsh-prog-notions\u002F3.boucles",{"title":203,"path":204,"stem":205},"Définitions et appels de fonctions","\u002Fsnt-2nde\u002Fsh-prog-notions\u002Ffonctions","0.snt-2nde\u002Fsh-prog-notions\u002F4.fonctions",{"title":207,"path":208,"stem":209,"children":210},"Démo des widgets MDC","\u002Fsnt-2nde\u002Fzz-demo-widgets","0.snt-2nde\u002Fzz-demo-widgets\u002F0.index",[211,212,216,220],{"title":207,"path":208,"stem":209},{"title":213,"path":214,"stem":215},"Démo statique — callouts, code, BO, illustration","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fstatique","0.snt-2nde\u002Fzz-demo-widgets\u002F1.statique",{"title":217,"path":218,"stem":219},"Démo interactive — self-eval et python-snippet","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Finteractif","0.snt-2nde\u002Fzz-demo-widgets\u002F2.interactif",{"title":221,"path":222,"stem":223},"Démo · 14 widgets de manipulation NSI","\u002Fsnt-2nde\u002Fzz-demo-widgets\u002Fwidgets-manipulation","0.snt-2nde\u002Fzz-demo-widgets\u002F3.widgets-manipulation",{"title":225,"path":226,"stem":227,"children":228},"NSI — 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deux","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fentiers-relatifs","1.nsi-1ere\u002Fpb-types-valeurs\u002F2.entiers-relatifs",{"title":271,"path":272,"stem":273},"Nombres flottants et virgule flottante","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fflottants","1.nsi-1ere\u002Fpb-types-valeurs\u002F3.flottants",{"title":275,"path":276,"stem":277},"Textes et encodages — ASCII, ISO-8859-1, Unicode","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Ftextes-encodages","1.nsi-1ere\u002Fpb-types-valeurs\u002F4.textes-encodages",{"title":279,"path":280,"stem":281},"Valeurs booléennes et opérateurs logiques","\u002Fnsi-1ere\u002Fpb-types-valeurs\u002Fbooleens","1.nsi-1ere\u002Fpb-types-valeurs\u002F5.booleens",{"title":283,"path":284,"stem":285,"children":286},"Représentation des données : types construits","\u002Fnsi-1ere\u002Fpc-types-construits","1.nsi-1ere\u002Fpc-types-construits\u002F0.index",[287,288,292,296,300],{"title":283,"path":284,"stem":285},{"title":289,"path":290,"stem":291},"Tableaux et listes","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftableaux-listes","1.nsi-1ere\u002Fpc-types-construits\u002F1.tableaux-listes",{"title":293,"path":294,"stem":295},"Listes en compréhension","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fcomprehension","1.nsi-1ere\u002Fpc-types-construits\u002F2.comprehension",{"title":297,"path":298,"stem":299},"Tuples et p-uplets nommés","\u002Fnsi-1ere\u002Fpc-types-construits\u002Ftuples","1.nsi-1ere\u002Fpc-types-construits\u002F3.tuples",{"title":301,"path":302,"stem":303},"Dictionnaires","\u002Fnsi-1ere\u002Fpc-types-construits\u002Fdictionnaires","1.nsi-1ere\u002Fpc-types-construits\u002F4.dictionnaires",{"title":305,"path":306,"stem":307,"children":308},"Traitement de données en tables","\u002Fnsi-1ere\u002Fpd-tables-donnees","1.nsi-1ere\u002Fpd-tables-donnees\u002F0.index",[309,310,314,318,322],{"title":305,"path":306,"stem":307},{"title":311,"path":312,"stem":313},"Tables et CSV","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftables-csv","1.nsi-1ere\u002Fpd-tables-donnees\u002F1.tables-csv",{"title":315,"path":316,"stem":317},"Recherche et filtrage","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Frecherche","1.nsi-1ere\u002Fpd-tables-donnees\u002F2.recherche",{"title":319,"path":320,"stem":321},"Trier une table","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ftri","1.nsi-1ere\u002Fpd-tables-donnees\u002F3.tri",{"title":323,"path":324,"stem":325,"children":326},"Fusion et indexation","\u002Fnsi-1ere\u002Fpd-tables-donnees\u002Ffusion-indexation","1.nsi-1ere\u002Fpd-tables-donnees\u002F4.fusion-indexation",[327],{"title":323,"path":324,"stem":325},{"title":329,"path":330,"stem":331,"children":332},"IHM sur le Web","\u002Fnsi-1ere\u002Fpe-ihm-web","1.nsi-1ere\u002Fpe-ihm-web\u002F0.index",[333,334,338,342,346,350],{"title":329,"path":330,"stem":331},{"title":335,"path":336,"stem":337},"HTML et CSS — rappel de structure","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fhtml-css-bases","1.nsi-1ere\u002Fpe-ihm-web\u002F1.html-css-bases",{"title":339,"path":340,"stem":341},"Formulaires HTML","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fformulaires","1.nsi-1ere\u002Fpe-ihm-web\u002F2.formulaires",{"title":343,"path":344,"stem":345},"Événements JavaScript","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fevenements-js","1.nsi-1ere\u002Fpe-ihm-web\u002F3.evenements-js",{"title":347,"path":348,"stem":349},"Manipulation du DOM","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fdom-interaction","1.nsi-1ere\u002Fpe-ihm-web\u002F4.dom-interaction",{"title":351,"path":352,"stem":353},"Client, serveur et HTTP","\u002Fnsi-1ere\u002Fpe-ihm-web\u002Fclient-serveur-http","1.nsi-1ere\u002Fpe-ihm-web\u002F5.client-serveur-http",{"title":355,"path":356,"stem":357,"children":358},"Architectures matérielles et systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os","1.nsi-1ere\u002Fpf-archi-os\u002F0.index",[359,360,364,368,372,376],{"title":355,"path":356,"stem":357},{"title":361,"path":362,"stem":363},"Le modèle de von Neumann","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fvon-neumann","1.nsi-1ere\u002Fpf-archi-os\u002F1.von-neumann",{"title":365,"path":366,"stem":367},"Périphériques et interface homme-machine","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fperipheriques-ihm","1.nsi-1ere\u002Fpf-archi-os\u002F2.peripheriques-ihm",{"title":369,"path":370,"stem":371},"Les systèmes d'exploitation","\u002Fnsi-1ere\u002Fpf-archi-os\u002Fsystemes-exploitation","1.nsi-1ere\u002Fpf-archi-os\u002F3.systemes-exploitation",{"title":373,"path":374,"stem":375},"Linux et le shell","\u002Fnsi-1ere\u002Fpf-archi-os\u002Flinux-shell","1.nsi-1ere\u002Fpf-archi-os\u002F4.linux-shell",{"title":377,"path":378,"stem":379},"Réseaux et protocoles","\u002Fnsi-1ere\u002Fpf-archi-os\u002Freseaux","1.nsi-1ere\u002Fpf-archi-os\u002F5.reseaux",{"title":381,"path":382,"stem":383,"children":384},"Langages et programmation","\u002Fnsi-1ere\u002Fpg-langages-prog","1.nsi-1ere\u002Fpg-langages-prog\u002F0.index",[385,386,390,394,398,402,406],{"title":381,"path":382,"stem":383},{"title":387,"path":388,"stem":389},"Diversité et unité des langages de programmation","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fdiversite-langages","1.nsi-1ere\u002Fpg-langages-prog\u002F1.diversite-langages",{"title":391,"path":392,"stem":393},"Constructions élémentaires","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fconstructions-elementaires","1.nsi-1ere\u002Fpg-langages-prog\u002F2.constructions-elementaires",{"title":395,"path":396,"stem":397},"Fonctions","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Ffonctions","1.nsi-1ere\u002Fpg-langages-prog\u002F3.fonctions",{"title":399,"path":400,"stem":401},"Spécification d'une fonction","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fspecification","1.nsi-1ere\u002Fpg-langages-prog\u002F4.specification",{"title":403,"path":404,"stem":405},"Mise au point des programmes","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fmise-au-point","1.nsi-1ere\u002Fpg-langages-prog\u002F5.mise-au-point",{"title":407,"path":408,"stem":409},"Utilisation de bibliothèques","\u002Fnsi-1ere\u002Fpg-langages-prog\u002Fbibliotheques","1.nsi-1ere\u002Fpg-langages-prog\u002F6.bibliotheques",{"title":411,"path":412,"stem":413,"children":414},"Algorithmique","\u002Fnsi-1ere\u002Fph-algorithmique","1.nsi-1ere\u002Fph-algorithmique\u002F0.index",[415,416,420,424,428,432,436],{"title":411,"path":412,"stem":413},{"title":417,"path":418,"stem":419},"Parcours séquentiel d'un tableau","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fparcours-sequentiel","1.nsi-1ere\u002Fph-algorithmique\u002F1.parcours-sequentiel",{"title":421,"path":422,"stem":423},"Bases de la complexité","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fcomplexite-bases","1.nsi-1ere\u002Fph-algorithmique\u002F2.complexite-bases",{"title":425,"path":426,"stem":427},"Tris par sélection et par insertion","\u002Fnsi-1ere\u002Fph-algorithmique\u002Ftris","1.nsi-1ere\u002Fph-algorithmique\u002F3.tris",{"title":429,"path":430,"stem":431},"Recherche dichotomique","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fdichotomie","1.nsi-1ere\u002Fph-algorithmique\u002F4.dichotomie",{"title":433,"path":434,"stem":435},"Algorithmes gloutons","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fgloutons","1.nsi-1ere\u002Fph-algorithmique\u002F5.gloutons",{"title":437,"path":438,"stem":439},"Les k plus proches voisins (k-NN)","\u002Fnsi-1ere\u002Fph-algorithmique\u002Fknn","1.nsi-1ere\u002Fph-algorithmique\u002F6.knn",{"title":441,"path":442,"stem":443,"children":444},"NSI — Terminale","\u002Fnsi-tale","2.nsi-tale\u002F0.index",[445,446,450,476,498,536,560],{"title":441,"path":442,"stem":443},{"title":447,"path":448,"stem":449},"Review pédagogique — chapitres NSI Terminale (TA, TB, TC, TD, TE)","\u002Fnsi-tale\u002Freview","2.nsi-tale\u002FREVIEW",{"title":451,"path":452,"stem":453,"children":454},"Structures de données","\u002Fnsi-tale\u002Fta-structures-donnees","2.nsi-tale\u002Fta-structures-donnees\u002F0.index",[455,456,460,464,468,472],{"title":451,"path":452,"stem":453},{"title":457,"path":458,"stem":459},"Objet, classe, interface","\u002Fnsi-tale\u002Fta-structures-donnees\u002Finterface-implementation","2.nsi-tale\u002Fta-structures-donnees\u002F1.interface-implementation",{"title":461,"path":462,"stem":463},"Implémenter une structure — l'exemple de la file FIFO","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fimplementer-une-structure","2.nsi-tale\u002Fta-structures-donnees\u002F2.implementer-une-structure",{"title":465,"path":466,"stem":467},"Listes, piles, files, dictionnaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Flistes-piles-files-dict","2.nsi-tale\u002Fta-structures-donnees\u002F3.listes-piles-files-dict",{"title":469,"path":470,"stem":471},"Arbres binaires","\u002Fnsi-tale\u002Fta-structures-donnees\u002Farbres-binaires","2.nsi-tale\u002Fta-structures-donnees\u002F4.arbres-binaires",{"title":473,"path":474,"stem":475},"Graphes","\u002Fnsi-tale\u002Fta-structures-donnees\u002Fgraphes","2.nsi-tale\u002Fta-structures-donnees\u002F5.graphes",{"title":477,"path":478,"stem":479,"children":480},"Bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees","2.nsi-tale\u002Ftb-bases-de-donnees\u002F0.index",[481,482,486,490,494],{"title":477,"path":478,"stem":479},{"title":483,"path":484,"stem":485},"Le modèle relationnel","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fmodele-relationnel","2.nsi-tale\u002Ftb-bases-de-donnees\u002F1.modele-relationnel",{"title":487,"path":488,"stem":489},"Bases de données relationnelles et anomalies de schéma","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fbase-relationnelle","2.nsi-tale\u002Ftb-bases-de-donnees\u002F2.base-relationnelle",{"title":491,"path":492,"stem":493},"Le langage SQL","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsql","2.nsi-tale\u002Ftb-bases-de-donnees\u002F3.sql",{"title":495,"path":496,"stem":497},"Les systèmes de gestion de bases de données","\u002Fnsi-tale\u002Ftb-bases-de-donnees\u002Fsgbd","2.nsi-tale\u002Ftb-bases-de-donnees\u002F4.sgbd",{"title":499,"path":500,"stem":501,"children":502},"Architectures matérielles, systèmes d'exploitation et réseaux","\u002Fnsi-tale\u002Ftc-archi-os-reseaux","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F0.index",[503,504,508,512,516,520,524,528,532],{"title":499,"path":500,"stem":501},{"title":505,"path":506,"stem":507},"Le système sur puce (SoC)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fsysteme-sur-puce","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F1.systeme-sur-puce",{"title":509,"path":510,"stem":511},"Processus et ordonnancement par l'OS","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprocessus-os","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F2.processus-os",{"title":513,"path":514,"stem":515},"L'interblocage (deadlock)","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Finterblocage","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F3.interblocage",{"title":517,"path":518,"stem":519},"Principes du routage","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Froutage-principes","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F4.routage-principes",{"title":521,"path":522,"stem":523},"Protocoles de routage RIP et OSPF","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fprotocoles-rip-ospf","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F5.protocoles-rip-ospf",{"title":525,"path":526,"stem":527},"Chiffrement symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-symetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F6.chiffrement-symetrique",{"title":529,"path":530,"stem":531},"Chiffrement asymétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fchiffrement-asymetrique","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F7.chiffrement-asymetrique",{"title":533,"path":534,"stem":535},"HTTPS — l'échange de clé symétrique","\u002Fnsi-tale\u002Ftc-archi-os-reseaux\u002Fhttps-tls","2.nsi-tale\u002Ftc-archi-os-reseaux\u002F8.https-tls",{"title":381,"path":537,"stem":538,"children":539},"\u002Fnsi-tale\u002Ftd-langages-prog","2.nsi-tale\u002Ftd-langages-prog\u002F0.index",[540,541,544,548,552,556],{"title":381,"path":537,"stem":538},{"title":403,"path":542,"stem":543},"\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmise-au-point","2.nsi-tale\u002Ftd-langages-prog\u002F1.mise-au-point",{"title":545,"path":546,"stem":547},"Modularité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fmodularite","2.nsi-tale\u002Ftd-langages-prog\u002F2.modularite",{"title":549,"path":550,"stem":551},"Récursivité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Frecursivite","2.nsi-tale\u002Ftd-langages-prog\u002F3.recursivite",{"title":553,"path":554,"stem":555},"Paradigmes de programmation","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fparadigmes","2.nsi-tale\u002Ftd-langages-prog\u002F4.paradigmes",{"title":557,"path":558,"stem":559},"Programme comme donnée — calculabilité et indécidabilité","\u002Fnsi-tale\u002Ftd-langages-prog\u002Fprogramme-comme-donnee","2.nsi-tale\u002Ftd-langages-prog\u002F5.programme-comme-donnee",{"title":411,"path":561,"stem":562,"children":563},"\u002Fnsi-tale\u002Fte-algorithmique","2.nsi-tale\u002Fte-algorithmique\u002F0.index",[564,565,569,573,577,581],{"title":411,"path":561,"stem":562},{"title":566,"path":567,"stem":568},"Algorithmes sur les arbres binaires","\u002Fnsi-tale\u002Fte-algorithmique\u002Farbres-binaires-algos","2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos",{"title":570,"path":571,"stem":572},"Parcours de graphes — DFS, BFS, cycles, chemins","\u002Fnsi-tale\u002Fte-algorithmique\u002Fgraphes-parcours","2.nsi-tale\u002Fte-algorithmique\u002F2.graphes-parcours",{"title":574,"path":575,"stem":576},"Diviser pour régner","\u002Fnsi-tale\u002Fte-algorithmique\u002Fdiviser-pour-regner","2.nsi-tale\u002Fte-algorithmique\u002F3.diviser-pour-regner",{"title":578,"path":579,"stem":580},"Programmation dynamique","\u002Fnsi-tale\u002Fte-algorithmique\u002Fprogrammation-dynamique","2.nsi-tale\u002Fte-algorithmique\u002F4.programmation-dynamique",{"title":582,"path":583,"stem":584},"Recherche textuelle — l'algorithme de Boyer-Moore","\u002Fnsi-tale\u002Fte-algorithmique\u002Frecherche-textuelle","2.nsi-tale\u002Fte-algorithmique\u002F5.recherche-textuelle",{"id":586,"title":566,"bo":587,"body":589,"description":2209,"eval":2210,"extension":2211,"labo":2210,"meta":2212,"navigation":751,"path":567,"quizzes":2213,"readingTime":665,"seo":2220,"stem":568,"tp":2210,"__hash__":2221},"cours\u002F2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos.md",[588],"TE01",{"type":590,"value":591,"toc":2200},"minimark",[592,597,610,625,634,637,641,652,852,856,867,978,1125,1147,1151,1162,1224,1389,1403,1407,1423,1570,1623,1644,1648,1779,1930,2076,2185,2189,2196],[593,594,596],"h2",{"id":595},"introduction","Introduction",[598,599,600,601,605,606,609],"p",{},"L'arbre binaire est une structure ",[602,603,604],"strong",{},"récursive par construction"," : chaque nœud\nporte une valeur et peut posséder un sous-arbre gauche et un sous-arbre droit,\neux-mêmes des arbres binaires. Cette nature récursive imprègne tous les\nalgorithmes qui s'y appliquent — ils suivent presque tous le schéma\n",[602,607,608],{},"« traiter la racine, recommencer à gauche, recommencer à droite »",".",[598,611,612,613,616,617,620,621,624],{},"Cette forme arborescente n'est pas une invention de l'informatique : on la\nretrouve dans la ",[602,614,615],{},"classification du vivant"," de Linné (règnes, ordres,\nespèces), dans un ",[602,618,619],{},"organigramme d'entreprise",", ou dans un ",[602,622,623],{},"arbre\ngénéalogique",". Partout où l'on décrit une hiérarchie qui se ramifie, on peut\nla lire comme un arbre.",[598,626,627,628,630,631,609],{},"Cette section suppose la familiarité avec la structure d'arbre binaire et son\nvocabulaire (racine, feuille, nœud interne, hauteur, taille) — voir le chapitre\n",[602,629,451],{}," (TA01). Ici, on les ",[602,632,633],{},"parcourt",[635,636],"bo-ref",{"code":588},[593,638,640],{"id":639},"une-classe-python-pour-les-nœuds","Une classe Python pour les nœuds",[598,642,643,644,648,649,609],{},"On représente un arbre par une classe ",[645,646,647],"code",{},"Noeud",", chaque nœud référençant ses\ndeux enfants éventuels. L'absence d'enfant est codée par ",[645,650,651],{},"None",[653,654,659],"pre",{"className":655,"code":656,"language":657,"meta":658,"style":658},"language-python shiki shiki-themes github-light github-light github-dark","class Noeud:\n    def __init__(self, valeur, gauche=None, droite=None):\n        self.valeur = valeur\n        self.gauche = gauche\n        self.droite = droite\n\n# Un petit arbre :\n#         4\n#        \u002F \\\n#       2   6\n#      \u002F \\   \\\n#     1   3   7\narbre = Noeud(4,\n              Noeud(2, Noeud(1), Noeud(3)),\n              Noeud(6, None, Noeud(7)))\n","python","",[645,660,661,678,706,720,733,746,753,760,766,772,778,784,790,807,831],{"__ignoreMap":658},[662,663,666,670,674],"span",{"class":664,"line":665},"line",1,[662,667,669],{"class":668},"s8jYJ","class",[662,671,673],{"class":672},"snPdu"," Noeud",[662,675,677],{"class":676},"sxrX7",":\n",[662,679,681,684,688,691,694,696,699,701,703],{"class":664,"line":680},2,[662,682,683],{"class":668},"    def",[662,685,687],{"class":686},"sBjJW"," __init__",[662,689,690],{"class":676},"(self, valeur, gauche",[662,692,693],{"class":668},"=",[662,695,651],{"class":686},[662,697,698],{"class":676},", droite",[662,700,693],{"class":668},[662,702,651],{"class":686},[662,704,705],{"class":676},"):\n",[662,707,709,712,715,717],{"class":664,"line":708},3,[662,710,711],{"class":686},"        self",[662,713,714],{"class":676},".valeur ",[662,716,693],{"class":668},[662,718,719],{"class":676}," valeur\n",[662,721,723,725,728,730],{"class":664,"line":722},4,[662,724,711],{"class":686},[662,726,727],{"class":676},".gauche ",[662,729,693],{"class":668},[662,731,732],{"class":676}," gauche\n",[662,734,736,738,741,743],{"class":664,"line":735},5,[662,737,711],{"class":686},[662,739,740],{"class":676},".droite ",[662,742,693],{"class":668},[662,744,745],{"class":676}," droite\n",[662,747,749],{"class":664,"line":748},6,[662,750,752],{"emptyLinePlaceholder":751},true,"\n",[662,754,756],{"class":664,"line":755},7,[662,757,759],{"class":758},"sCsY4","# Un petit arbre :\n",[662,761,763],{"class":664,"line":762},8,[662,764,765],{"class":758},"#         4\n",[662,767,769],{"class":664,"line":768},9,[662,770,771],{"class":758},"#        \u002F \\\n",[662,773,775],{"class":664,"line":774},10,[662,776,777],{"class":758},"#       2   6\n",[662,779,781],{"class":664,"line":780},11,[662,782,783],{"class":758},"#      \u002F \\   \\\n",[662,785,787],{"class":664,"line":786},12,[662,788,789],{"class":758},"#     1   3   7\n",[662,791,793,796,798,801,804],{"class":664,"line":792},13,[662,794,795],{"class":676},"arbre ",[662,797,693],{"class":668},[662,799,800],{"class":676}," Noeud(",[662,802,803],{"class":686},"4",[662,805,806],{"class":676},",\n",[662,808,810,813,816,819,822,825,828],{"class":664,"line":809},14,[662,811,812],{"class":676},"              Noeud(",[662,814,815],{"class":686},"2",[662,817,818],{"class":676},", Noeud(",[662,820,821],{"class":686},"1",[662,823,824],{"class":676},"), Noeud(",[662,826,827],{"class":686},"3",[662,829,830],{"class":676},")),\n",[662,832,834,836,839,842,844,846,849],{"class":664,"line":833},15,[662,835,812],{"class":676},[662,837,838],{"class":686},"6",[662,840,841],{"class":676},", ",[662,843,651],{"class":686},[662,845,818],{"class":676},[662,847,848],{"class":686},"7",[662,850,851],{"class":676},")))\n",[593,853,855],{"id":854},"taille-et-hauteur","Taille et hauteur",[598,857,858,859,862,863,866],{},"La ",[602,860,861],{},"taille"," d'un arbre est son nombre de nœuds ; sa ",[602,864,865],{},"hauteur"," est la longueur\ndu plus long chemin de la racine à une feuille. Toutes deux se calculent par\nrécurrence en suivant le squelette de la structure.",[653,868,870],{"className":655,"code":869,"language":657,"meta":658,"style":658},"def taille(a):\n    if a is None:\n        return 0\n    return 1 + taille(a.gauche) + taille(a.droite)\n\ndef hauteur(a):\n    if a is None:\n        return -1                          # par convention : arbre vide\n    return 1 + max(hauteur(a.gauche), hauteur(a.droite))\n",[645,871,872,883,899,907,927,931,940,952,964],{"__ignoreMap":658},[662,873,874,877,880],{"class":664,"line":665},[662,875,876],{"class":668},"def",[662,878,879],{"class":672}," taille",[662,881,882],{"class":676},"(a):\n",[662,884,885,888,891,894,897],{"class":664,"line":680},[662,886,887],{"class":668},"    if",[662,889,890],{"class":676}," a ",[662,892,893],{"class":668},"is",[662,895,896],{"class":686}," None",[662,898,677],{"class":676},[662,900,901,904],{"class":664,"line":708},[662,902,903],{"class":668},"        return",[662,905,906],{"class":686}," 0\n",[662,908,909,912,915,918,921,924],{"class":664,"line":722},[662,910,911],{"class":668},"    return",[662,913,914],{"class":686}," 1",[662,916,917],{"class":668}," +",[662,919,920],{"class":676}," taille(a.gauche) ",[662,922,923],{"class":668},"+",[662,925,926],{"class":676}," taille(a.droite)\n",[662,928,929],{"class":664,"line":735},[662,930,752],{"emptyLinePlaceholder":751},[662,932,933,935,938],{"class":664,"line":748},[662,934,876],{"class":668},[662,936,937],{"class":672}," hauteur",[662,939,882],{"class":676},[662,941,942,944,946,948,950],{"class":664,"line":755},[662,943,887],{"class":668},[662,945,890],{"class":676},[662,947,893],{"class":668},[662,949,896],{"class":686},[662,951,677],{"class":676},[662,953,954,956,959,961],{"class":664,"line":762},[662,955,903],{"class":668},[662,957,958],{"class":668}," -",[662,960,821],{"class":686},[662,962,963],{"class":758},"                          # par convention : arbre vide\n",[662,965,966,968,970,972,975],{"class":664,"line":768},[662,967,911],{"class":668},[662,969,914],{"class":686},[662,971,917],{"class":668},[662,973,974],{"class":686}," max",[662,976,977],{"class":676},"(hauteur(a.gauche), hauteur(a.droite))\n",[598,979,980,983,984,1035,1036,1066,1067,1124],{},[602,981,982],{},"Convention"," : la hauteur de l'arbre vide vaut ",[662,985,988,1014],{"className":986},[987],"katex",[662,989,992],{"className":990},[991],"katex-mathml",[993,994,996],"math",{"xmlns":995},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[997,998,999,1009],"semantics",{},[1000,1001,1002,1006],"mrow",{},[1003,1004,1005],"mo",{},"−",[1007,1008,821],"mn",{},[1010,1011,1013],"annotation",{"encoding":1012},"application\u002Fx-tex","-1",[662,1015,1019],{"className":1016,"ariaHidden":1018},[1017],"katex-html","true",[662,1020,1023,1028,1032],{"className":1021},[1022],"base",[662,1024],{"className":1025,"style":1027},[1026],"strut","height:0.7278em;vertical-align:-0.0833em;",[662,1029,1005],{"className":1030},[1031],"mord",[662,1033,821],{"className":1034},[1031],", celle d'une feuille vaut\n",[662,1037,1039,1053],{"className":1038},[987],[662,1040,1042],{"className":1041},[991],[993,1043,1044],{"xmlns":995},[997,1045,1046,1051],{},[1000,1047,1048],{},[1007,1049,1050],{},"0",[1010,1052,1050],{"encoding":1012},[662,1054,1056],{"className":1055,"ariaHidden":1018},[1017],[662,1057,1059,1063],{"className":1058},[1022],[662,1060],{"className":1061,"style":1062},[1026],"height:0.6444em;",[662,1064,1050],{"className":1065},[1031],". Le coût des deux fonctions est en ",[662,1068,1070,1097],{"className":1069},[987],[662,1071,1073],{"className":1072},[991],[993,1074,1075],{"xmlns":995},[997,1076,1077,1094],{},[1000,1078,1079,1084,1088,1091],{},[1080,1081,1083],"mi",{"mathvariant":1082},"script","O",[1003,1085,1087],{"stretchy":1086},"false","(",[1080,1089,1090],{},"n",[1003,1092,1093],{"stretchy":1086},")",[1010,1095,1096],{"encoding":1012},"\\mathcal{O}(n)",[662,1098,1100],{"className":1099,"ariaHidden":1018},[1017],[662,1101,1103,1107,1112,1116,1120],{"className":1102},[1022],[662,1104],{"className":1105,"style":1106},[1026],"height:1em;vertical-align:-0.25em;",[662,1108,1083],{"className":1109,"style":1111},[1031,1110],"mathcal","margin-right:0.0278em;",[662,1113,1087],{"className":1114},[1115],"mopen",[662,1117,1090],{"className":1118},[1031,1119],"mathnormal",[662,1121,1093],{"className":1122},[1123],"mclose"," — chaque nœud est visité\nexactement une fois.",[1126,1127,1129,1135],"self-eval-qcm",{"correct":815,"slug":1128},"arbres-binaires-algos-c1",[1130,1131,1132],"template",{"v-slot:question":658},[598,1133,1134],{},"Quelle est la hauteur de l'arbre dessiné ci-dessus ?",[1130,1136,1137],{"v-slot:choices":658},[1138,1139,1140,1143,1145],"ul",{},[1141,1142,821],"li",{},[1141,1144,815],{},[1141,1146,827],{},[593,1148,1150],{"id":1149},"les-trois-parcours-en-profondeur","Les trois parcours en profondeur",[598,1152,1153,1154,1157,1158,1161],{},"Un parcours en ",[602,1155,1156],{},"profondeur"," visite récursivement un sous-arbre entier avant\nde passer au suivant. Trois ordres se distinguent selon ",[602,1159,1160],{},"quand"," on traite la\nracine relativement à ses enfants.",[1163,1164,1165,1181],"table",{},[1166,1167,1168],"thead",{},[1169,1170,1171,1175,1178],"tr",{},[1172,1173,1174],"th",{},"Parcours",[1172,1176,1177],{},"Ordre",[1172,1179,1180],{},"Sur l'arbre exemple",[1182,1183,1184,1198,1211],"tbody",{},[1169,1185,1186,1192,1195],{},[1187,1188,1189],"td",{},[602,1190,1191],{},"Préfixe",[1187,1193,1194],{},"racine → gauche → droite",[1187,1196,1197],{},"4, 2, 1, 3, 6, 7",[1169,1199,1200,1205,1208],{},[1187,1201,1202],{},[602,1203,1204],{},"Infixe",[1187,1206,1207],{},"gauche → racine → droite",[1187,1209,1210],{},"1, 2, 3, 4, 6, 7",[1169,1212,1213,1218,1221],{},[1187,1214,1215],{},[602,1216,1217],{},"Suffixe",[1187,1219,1220],{},"gauche → droite → racine",[1187,1222,1223],{},"1, 3, 2, 7, 6, 4",[653,1225,1227],{"className":655,"code":1226,"language":657,"meta":658,"style":658},"def prefixe(a):\n    if a is None: return\n    print(a.valeur, end=\" \")\n    prefixe(a.gauche)\n    prefixe(a.droite)\n\ndef infixe(a):\n    if a is None: return\n    infixe(a.gauche)\n    print(a.valeur, end=\" \")\n    infixe(a.droite)\n\ndef suffixe(a):\n    if a is None: return\n    suffixe(a.gauche)\n    suffixe(a.droite)\n    print(a.valeur, end=\" \")\n",[645,1228,1229,1238,1254,1275,1280,1285,1289,1298,1312,1317,1331,1336,1340,1349,1363,1368,1374],{"__ignoreMap":658},[662,1230,1231,1233,1236],{"class":664,"line":665},[662,1232,876],{"class":668},[662,1234,1235],{"class":672}," prefixe",[662,1237,882],{"class":676},[662,1239,1240,1242,1244,1246,1248,1251],{"class":664,"line":680},[662,1241,887],{"class":668},[662,1243,890],{"class":676},[662,1245,893],{"class":668},[662,1247,896],{"class":686},[662,1249,1250],{"class":676},": ",[662,1252,1253],{"class":668},"return\n",[662,1255,1256,1259,1262,1266,1268,1272],{"class":664,"line":708},[662,1257,1258],{"class":686},"    print",[662,1260,1261],{"class":676},"(a.valeur, ",[662,1263,1265],{"class":1264},"sP4rz","end",[662,1267,693],{"class":668},[662,1269,1271],{"class":1270},"sIIMD","\" \"",[662,1273,1274],{"class":676},")\n",[662,1276,1277],{"class":664,"line":722},[662,1278,1279],{"class":676},"    prefixe(a.gauche)\n",[662,1281,1282],{"class":664,"line":735},[662,1283,1284],{"class":676},"    prefixe(a.droite)\n",[662,1286,1287],{"class":664,"line":748},[662,1288,752],{"emptyLinePlaceholder":751},[662,1290,1291,1293,1296],{"class":664,"line":755},[662,1292,876],{"class":668},[662,1294,1295],{"class":672}," infixe",[662,1297,882],{"class":676},[662,1299,1300,1302,1304,1306,1308,1310],{"class":664,"line":762},[662,1301,887],{"class":668},[662,1303,890],{"class":676},[662,1305,893],{"class":668},[662,1307,896],{"class":686},[662,1309,1250],{"class":676},[662,1311,1253],{"class":668},[662,1313,1314],{"class":664,"line":768},[662,1315,1316],{"class":676},"    infixe(a.gauche)\n",[662,1318,1319,1321,1323,1325,1327,1329],{"class":664,"line":774},[662,1320,1258],{"class":686},[662,1322,1261],{"class":676},[662,1324,1265],{"class":1264},[662,1326,693],{"class":668},[662,1328,1271],{"class":1270},[662,1330,1274],{"class":676},[662,1332,1333],{"class":664,"line":780},[662,1334,1335],{"class":676},"    infixe(a.droite)\n",[662,1337,1338],{"class":664,"line":786},[662,1339,752],{"emptyLinePlaceholder":751},[662,1341,1342,1344,1347],{"class":664,"line":792},[662,1343,876],{"class":668},[662,1345,1346],{"class":672}," suffixe",[662,1348,882],{"class":676},[662,1350,1351,1353,1355,1357,1359,1361],{"class":664,"line":809},[662,1352,887],{"class":668},[662,1354,890],{"class":676},[662,1356,893],{"class":668},[662,1358,896],{"class":686},[662,1360,1250],{"class":676},[662,1362,1253],{"class":668},[662,1364,1365],{"class":664,"line":833},[662,1366,1367],{"class":676},"    suffixe(a.gauche)\n",[662,1369,1371],{"class":664,"line":1370},16,[662,1372,1373],{"class":676},"    suffixe(a.droite)\n",[662,1375,1377,1379,1381,1383,1385,1387],{"class":664,"line":1376},17,[662,1378,1258],{"class":686},[662,1380,1261],{"class":676},[662,1382,1265],{"class":1264},[662,1384,693],{"class":668},[662,1386,1271],{"class":1270},[662,1388,1274],{"class":676},[1390,1391,1392],"note",{},[598,1393,1394,1395,1398,1399,1402],{},"Sur un ",[602,1396,1397],{},"arbre binaire de recherche",", le parcours infixe affiche les valeurs\ndans ",[602,1400,1401],{},"l'ordre croissant",". C'est une propriété structurelle, exploitée pour\ntrier des données ou valider qu'un arbre respecte bien la propriété d'ABR.",[593,1404,1406],{"id":1405},"parcours-en-largeur-bfs","Parcours en largeur (BFS)",[598,1408,1409,1410,1413,1414,1418,1419,1422],{},"Le parcours ",[602,1411,1412],{},"en largeur"," (BFS, ",[1415,1416,1417],"em",{},"Breadth-First Search",") visite les nœuds\nniveau par niveau. Il ne se programme pas naturellement par récursion : on\nutilise une ",[602,1420,1421],{},"file"," (FIFO).",[653,1424,1426],{"className":655,"code":1425,"language":657,"meta":658,"style":658},"from collections import deque\n\ndef parcours_largeur(a):\n    if a is None: return\n    file = deque([a])\n    while file:\n        n = file.popleft()\n        print(n.valeur, end=\" \")\n        if n.gauche is not None:  file.append(n.gauche)\n        if n.droite is not None:  file.append(n.droite)\n\n# Sortie sur l'arbre exemple : 4 2 6 1 3 7\n",[645,1427,1428,1442,1446,1455,1469,1480,1490,1502,1518,1541,1561,1565],{"__ignoreMap":658},[662,1429,1430,1433,1436,1439],{"class":664,"line":665},[662,1431,1432],{"class":668},"from",[662,1434,1435],{"class":676}," collections ",[662,1437,1438],{"class":668},"import",[662,1440,1441],{"class":676}," deque\n",[662,1443,1444],{"class":664,"line":680},[662,1445,752],{"emptyLinePlaceholder":751},[662,1447,1448,1450,1453],{"class":664,"line":708},[662,1449,876],{"class":668},[662,1451,1452],{"class":672}," parcours_largeur",[662,1454,882],{"class":676},[662,1456,1457,1459,1461,1463,1465,1467],{"class":664,"line":722},[662,1458,887],{"class":668},[662,1460,890],{"class":676},[662,1462,893],{"class":668},[662,1464,896],{"class":686},[662,1466,1250],{"class":676},[662,1468,1253],{"class":668},[662,1470,1471,1474,1477],{"class":664,"line":735},[662,1472,1473],{"class":1264},"    file",[662,1475,1476],{"class":668}," =",[662,1478,1479],{"class":676}," deque([a])\n",[662,1481,1482,1485,1488],{"class":664,"line":748},[662,1483,1484],{"class":668},"    while",[662,1486,1487],{"class":1264}," file",[662,1489,677],{"class":676},[662,1491,1492,1495,1497,1499],{"class":664,"line":755},[662,1493,1494],{"class":676},"        n ",[662,1496,693],{"class":668},[662,1498,1487],{"class":1264},[662,1500,1501],{"class":676},".popleft()\n",[662,1503,1504,1507,1510,1512,1514,1516],{"class":664,"line":762},[662,1505,1506],{"class":686},"        print",[662,1508,1509],{"class":676},"(n.valeur, ",[662,1511,1265],{"class":1264},[662,1513,693],{"class":668},[662,1515,1271],{"class":1270},[662,1517,1274],{"class":676},[662,1519,1520,1523,1526,1528,1531,1533,1536,1538],{"class":664,"line":768},[662,1521,1522],{"class":668},"        if",[662,1524,1525],{"class":676}," n.gauche ",[662,1527,893],{"class":668},[662,1529,1530],{"class":668}," not",[662,1532,896],{"class":686},[662,1534,1535],{"class":676},":  ",[662,1537,1421],{"class":1264},[662,1539,1540],{"class":676},".append(n.gauche)\n",[662,1542,1543,1545,1548,1550,1552,1554,1556,1558],{"class":664,"line":774},[662,1544,1522],{"class":668},[662,1546,1547],{"class":676}," n.droite ",[662,1549,893],{"class":668},[662,1551,1530],{"class":668},[662,1553,896],{"class":686},[662,1555,1535],{"class":676},[662,1557,1421],{"class":1264},[662,1559,1560],{"class":676},".append(n.droite)\n",[662,1562,1563],{"class":664,"line":780},[662,1564,752],{"emptyLinePlaceholder":751},[662,1566,1567],{"class":664,"line":786},[662,1568,1569],{"class":758},"# Sortie sur l'arbre exemple : 4 2 6 1 3 7\n",[598,1571,1572,1575,1576,1579,1580,609],{},[602,1573,1574],{},"Invariant de boucle"," : à chaque itération, la file contient les nœuds\n",[602,1577,1578],{},"découverts mais non encore traités",", du moins profond au plus profond.\nQuand la file se vide, tout l'arbre a été parcouru — d'où le coût en\n",[662,1581,1583,1602],{"className":1582},[987],[662,1584,1586],{"className":1585},[991],[993,1587,1588],{"xmlns":995},[997,1589,1590,1600],{},[1000,1591,1592,1594,1596,1598],{},[1080,1593,1083],{"mathvariant":1082},[1003,1595,1087],{"stretchy":1086},[1080,1597,1090],{},[1003,1599,1093],{"stretchy":1086},[1010,1601,1096],{"encoding":1012},[662,1603,1605],{"className":1604,"ariaHidden":1018},[1017],[662,1606,1608,1611,1614,1617,1620],{"className":1607},[1022],[662,1609],{"className":1610,"style":1106},[1026],[662,1612,1083],{"className":1613,"style":1111},[1031,1110],[662,1615,1087],{"className":1616},[1115],[662,1618,1090],{"className":1619},[1031,1119],[662,1621,1093],{"className":1622},[1123],[1126,1624,1626,1631],{"correct":821,"slug":1625},"arbres-binaires-algos-c2",[1130,1627,1628],{"v-slot:question":658},[598,1629,1630],{},"Quelle structure de données est essentielle au parcours en largeur ?",[1130,1632,1633],{"v-slot:choices":658},[1138,1634,1635,1638,1641],{},[1141,1636,1637],{},"une pile (LIFO)",[1141,1639,1640],{},"une file (FIFO)",[1141,1642,1643],{},"un dictionnaire",[593,1645,1647],{"id":1646},"arbre-binaire-de-recherche-rechercher-insérer","Arbre binaire de recherche : rechercher, insérer",[598,1649,1650,1651,1653,1654,1685,1686,1734,1735,609],{},"Un ",[602,1652,1397],{}," (ABR) respecte une propriété d'ordre : pour\ntout nœud de valeur ",[662,1655,1657,1671],{"className":1656},[987],[662,1658,1660],{"className":1659},[991],[993,1661,1662],{"xmlns":995},[997,1663,1664,1669],{},[1000,1665,1666],{},[1080,1667,1668],{},"v",[1010,1670,1668],{"encoding":1012},[662,1672,1674],{"className":1673,"ariaHidden":1018},[1017],[662,1675,1677,1681],{"className":1676},[1022],[662,1678],{"className":1679,"style":1680},[1026],"height:0.4306em;",[662,1682,1668],{"className":1683,"style":1684},[1031,1119],"margin-right:0.0359em;",", toutes les valeurs du sous-arbre gauche sont ",[662,1687,1689,1706],{"className":1688},[987],[662,1690,1692],{"className":1691},[991],[993,1693,1694],{"xmlns":995},[997,1695,1696,1703],{},[1000,1697,1698,1701],{},[1003,1699,1700],{},"\u003C",[1080,1702,1668],{},[1010,1704,1705],{"encoding":1012},"\u003C v",[662,1707,1709,1725],{"className":1708,"ariaHidden":1018},[1017],[662,1710,1712,1716,1720],{"className":1711},[1022],[662,1713],{"className":1714,"style":1715},[1026],"height:0.5782em;vertical-align:-0.0391em;",[662,1717,1700],{"className":1718},[1719],"mrel",[662,1721],{"className":1722,"style":1724},[1723],"mspace","margin-right:0.2778em;",[662,1726,1728,1731],{"className":1727},[1022],[662,1729],{"className":1730,"style":1680},[1026],[662,1732,1668],{"className":1733,"style":1684},[1031,1119],",\net toutes celles du sous-arbre droit sont ",[662,1736,1738,1755],{"className":1737},[987],[662,1739,1741],{"className":1740},[991],[993,1742,1743],{"xmlns":995},[997,1744,1745,1752],{},[1000,1746,1747,1750],{},[1003,1748,1749],{},">",[1080,1751,1668],{},[1010,1753,1754],{"encoding":1012},"> v",[662,1756,1758,1770],{"className":1757,"ariaHidden":1018},[1017],[662,1759,1761,1764,1767],{"className":1760},[1022],[662,1762],{"className":1763,"style":1715},[1026],[662,1765,1749],{"className":1766},[1719],[662,1768],{"className":1769,"style":1724},[1723],[662,1771,1773,1776],{"className":1772},[1022],[662,1774],{"className":1775,"style":1680},[1026],[662,1777,1668],{"className":1778,"style":1684},[1031,1119],[653,1780,1782],{"className":655,"code":1781,"language":657,"meta":658,"style":658},"def rechercher(a, cle):\n    if a is None:           return False\n    if cle == a.valeur:     return True\n    if cle \u003C  a.valeur:     return rechercher(a.gauche, cle)\n    return rechercher(a.droite, cle)\n\ndef inserer(a, cle):\n    if a is None:                       return Noeud(cle)\n    if cle \u003C  a.valeur:  a.gauche = inserer(a.gauche, cle)\n    elif cle > a.valeur: a.droite = inserer(a.droite, cle)\n    # Si cle == a.valeur, on ne fait rien : pas de doublons dans cet ABR.\n    return a\n",[645,1783,1784,1794,1813,1831,1847,1854,1858,1867,1885,1901,1918,1923],{"__ignoreMap":658},[662,1785,1786,1788,1791],{"class":664,"line":665},[662,1787,876],{"class":668},[662,1789,1790],{"class":672}," rechercher",[662,1792,1793],{"class":676},"(a, cle):\n",[662,1795,1796,1798,1800,1802,1804,1807,1810],{"class":664,"line":680},[662,1797,887],{"class":668},[662,1799,890],{"class":676},[662,1801,893],{"class":668},[662,1803,896],{"class":686},[662,1805,1806],{"class":676},":           ",[662,1808,1809],{"class":668},"return",[662,1811,1812],{"class":686}," False\n",[662,1814,1815,1817,1820,1823,1826,1828],{"class":664,"line":708},[662,1816,887],{"class":668},[662,1818,1819],{"class":676}," cle ",[662,1821,1822],{"class":668},"==",[662,1824,1825],{"class":676}," a.valeur:     ",[662,1827,1809],{"class":668},[662,1829,1830],{"class":686}," True\n",[662,1832,1833,1835,1837,1839,1842,1844],{"class":664,"line":722},[662,1834,887],{"class":668},[662,1836,1819],{"class":676},[662,1838,1700],{"class":668},[662,1840,1841],{"class":676},"  a.valeur:     ",[662,1843,1809],{"class":668},[662,1845,1846],{"class":676}," rechercher(a.gauche, cle)\n",[662,1848,1849,1851],{"class":664,"line":735},[662,1850,911],{"class":668},[662,1852,1853],{"class":676}," rechercher(a.droite, cle)\n",[662,1855,1856],{"class":664,"line":748},[662,1857,752],{"emptyLinePlaceholder":751},[662,1859,1860,1862,1865],{"class":664,"line":755},[662,1861,876],{"class":668},[662,1863,1864],{"class":672}," inserer",[662,1866,1793],{"class":676},[662,1868,1869,1871,1873,1875,1877,1880,1882],{"class":664,"line":762},[662,1870,887],{"class":668},[662,1872,890],{"class":676},[662,1874,893],{"class":668},[662,1876,896],{"class":686},[662,1878,1879],{"class":676},":                       ",[662,1881,1809],{"class":668},[662,1883,1884],{"class":676}," Noeud(cle)\n",[662,1886,1887,1889,1891,1893,1896,1898],{"class":664,"line":768},[662,1888,887],{"class":668},[662,1890,1819],{"class":676},[662,1892,1700],{"class":668},[662,1894,1895],{"class":676},"  a.valeur:  a.gauche ",[662,1897,693],{"class":668},[662,1899,1900],{"class":676}," inserer(a.gauche, cle)\n",[662,1902,1903,1906,1908,1910,1913,1915],{"class":664,"line":774},[662,1904,1905],{"class":668},"    elif",[662,1907,1819],{"class":676},[662,1909,1749],{"class":668},[662,1911,1912],{"class":676}," a.valeur: a.droite ",[662,1914,693],{"class":668},[662,1916,1917],{"class":676}," inserer(a.droite, cle)\n",[662,1919,1920],{"class":664,"line":780},[662,1921,1922],{"class":758},"    # Si cle == a.valeur, on ne fait rien : pas de doublons dans cet ABR.\n",[662,1924,1925,1927],{"class":664,"line":786},[662,1926,911],{"class":668},[662,1928,1929],{"class":676}," a\n",[598,1931,1932,1933,1936,1937,1940,1941,2006,2007,2010,2011,2014,2015,2071,2072,2075],{},"À chaque étape, on ",[602,1934,1935],{},"élimine la moitié"," des descendants encore à explorer.\nSi l'arbre est ",[602,1938,1939],{},"équilibré"," (hauteur ",[662,1942,1944,1972],{"className":1943},[987],[662,1945,1947],{"className":1946},[991],[993,1948,1949],{"xmlns":995},[997,1950,1951,1969],{},[1000,1952,1953,1957,1959,1962,1965,1967],{},[1080,1954,1956],{"mathvariant":1955},"normal","Θ",[1003,1958,1087],{"stretchy":1086},[1080,1960,1961],{},"log",[1003,1963,1964],{},"⁡",[1080,1966,1090],{},[1003,1968,1093],{"stretchy":1086},[1010,1970,1971],{"encoding":1012},"\\Theta(\\log n)",[662,1973,1975],{"className":1974,"ariaHidden":1018},[1017],[662,1976,1978,1981,1984,1987,1996,2000,2003],{"className":1977},[1022],[662,1979],{"className":1980,"style":1106},[1026],[662,1982,1956],{"className":1983},[1031],[662,1985,1087],{"className":1986},[1115],[662,1988,1991,1992],{"className":1989},[1990],"mop","lo",[662,1993,1995],{"style":1994},"margin-right:0.0139em;","g",[662,1997],{"className":1998,"style":1999},[1723],"margin-right:0.1667em;",[662,2001,1090],{"className":2002},[1031,1119],[662,2004,1093],{"className":2005},[1123],"), le coût de\n",[645,2008,2009],{},"rechercher"," et ",[645,2012,2013],{},"inserer"," est en ",[662,2016,2018,2042],{"className":2017},[987],[662,2019,2021],{"className":2020},[991],[993,2022,2023],{"xmlns":995},[997,2024,2025,2039],{},[1000,2026,2027,2029,2031,2033,2035,2037],{},[1080,2028,1083],{"mathvariant":1082},[1003,2030,1087],{"stretchy":1086},[1080,2032,1961],{},[1003,2034,1964],{},[1080,2036,1090],{},[1003,2038,1093],{"stretchy":1086},[1010,2040,2041],{"encoding":1012},"\\mathcal{O}(\\log n)",[662,2043,2045],{"className":2044,"ariaHidden":1018},[1017],[662,2046,2048,2051,2054,2057,2062,2065,2068],{"className":2047},[1022],[662,2049],{"className":2050,"style":1106},[1026],[662,2052,1083],{"className":2053,"style":1111},[1031,1110],[662,2055,1087],{"className":2056},[1115],[662,2058,1991,2060],{"className":2059},[1990],[662,2061,1995],{"style":1994},[662,2063],{"className":2064,"style":1999},[1723],[662,2066,1090],{"className":2067},[1031,1119],[662,2069,1093],{"className":2070},[1123]," — c'est l'intérêt\nmajeur de la structure. On peut voir un ABR équilibré comme un ",[602,2073,2074],{},"annuaire\ntrié vivant"," : la recherche d'un nom y obéit au même principe que la\nrecherche dichotomique dans un dictionnaire papier — on ouvre au milieu,\npuis on choisit la moitié gauche ou droite selon l'ordre alphabétique.",[2077,2078,2079],"warning",{},[598,2080,2081,2082,2085,2086,2140,2141,2184],{},"Si l'arbre est ",[602,2083,2084],{},"dégénéré"," (par exemple en insérant des clés déjà triées), il\ndevient une simple liste chaînée de hauteur ",[662,2087,2089,2107],{"className":2088},[987],[662,2090,2092],{"className":2091},[991],[993,2093,2094],{"xmlns":995},[997,2095,2096,2104],{},[1000,2097,2098,2100,2102],{},[1080,2099,1090],{},[1003,2101,1005],{},[1007,2103,821],{},[1010,2105,2106],{"encoding":1012},"n-1",[662,2108,2110,2131],{"className":2109,"ariaHidden":1018},[1017],[662,2111,2113,2117,2120,2124,2128],{"className":2112},[1022],[662,2114],{"className":2115,"style":2116},[1026],"height:0.6667em;vertical-align:-0.0833em;",[662,2118,1090],{"className":2119},[1031,1119],[662,2121],{"className":2122,"style":2123},[1723],"margin-right:0.2222em;",[662,2125,1005],{"className":2126},[2127],"mbin",[662,2129],{"className":2130,"style":2123},[1723],[662,2132,2134,2137],{"className":2133},[1022],[662,2135],{"className":2136,"style":1062},[1026],[662,2138,821],{"className":2139},[1031],", et le coût retombe à\n",[662,2142,2144,2163],{"className":2143},[987],[662,2145,2147],{"className":2146},[991],[993,2148,2149],{"xmlns":995},[997,2150,2151,2161],{},[1000,2152,2153,2155,2157,2159],{},[1080,2154,1083],{"mathvariant":1082},[1003,2156,1087],{"stretchy":1086},[1080,2158,1090],{},[1003,2160,1093],{"stretchy":1086},[1010,2162,1096],{"encoding":1012},[662,2164,2166],{"className":2165,"ariaHidden":1018},[1017],[662,2167,2169,2172,2175,2178,2181],{"className":2168},[1022],[662,2170],{"className":2171,"style":1106},[1026],[662,2173,1083],{"className":2174,"style":1111},[1031,1110],[662,2176,1087],{"className":2177},[1115],[662,2179,1090],{"className":2180},[1031,1119],[662,2182,1093],{"className":2183},[1123],". Les ABR équilibrés (AVL, rouge-noir) corrigent cela ; au\nbaccalauréat on suppose simplement que l'arbre est « raisonnablement\néquilibré ».",[593,2186,2188],{"id":2187},"pour-aller-plus-loin","Pour aller plus loin",[598,2190,2191,2192,2195],{},"La récursion sur les arbres se prête particulièrement bien à la ",[602,2193,2194],{},"preuve par\ninduction structurelle"," : on démontre une propriété pour l'arbre vide (cas de\nbase), puis pour un nœud à partir de l'hypothèse sur ses deux sous-arbres\n(hérédité). C'est la version arborescente du raisonnement par récurrence.",[2197,2198,2199],"style",{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html pre.shiki code .sCsY4, html code.shiki .sCsY4{--shiki-light:#6A737D;--shiki-default:#6A737D;--shiki-dark:#6A737D}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: 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équilibré.",null,"md",{},[2214,2218],{"slug":1128,"kind":2215,"question":1134,"choices":2216,"correct":815,"multi":2217,"explanation":658},"qcm",[821,815,827],false,{"slug":1625,"kind":2215,"question":1630,"choices":2219,"correct":821,"multi":2217,"explanation":658},[1637,1640,1643],{"title":566,"description":2209},"Dd6IS5q6738tlpeJh2MqXmK4Y091W3287CvvWGoRtYk",{"id":2223,"title":2224,"bo":2225,"body":2226,"description":3280,"eval":2210,"extension":2211,"labo":2210,"meta":3281,"navigation":2217,"path":3282,"quizzes":3283,"readingTime":665,"seo":3284,"stem":3285,"tp":2210,"__hash__":3286},"coursMemo\u002F2.nsi-tale\u002Fte-algorithmique\u002F1.arbres-binaires-algos.memo.md","Algorithmes sur les arbres binaires — l'essentiel",[588],{"type":590,"value":2227,"toc":3272},[2228,2232,2295,2299,2925,2929,2987,2991,3097,3101,3171,3175,3269],[593,2229,2231],{"id":2230},"en-une-phrase","En une phrase",[598,2233,2234,2235,2238,2239,2294],{},"Les algorithmes d'arbres suivent la ",[602,2236,2237],{},"récursion de la structure"," ; les trois\nparcours en profondeur diffèrent par la position de la racine ; le BFS utilise\nune file ; l'ABR offre une recherche en ",[662,2240,2242,2265],{"className":2241},[987],[662,2243,2245],{"className":2244},[991],[993,2246,2247],{"xmlns":995},[997,2248,2249,2263],{},[1000,2250,2251,2253,2255,2257,2259,2261],{},[1080,2252,1083],{"mathvariant":1082},[1003,2254,1087],{"stretchy":1086},[1080,2256,1961],{},[1003,2258,1964],{},[1080,2260,1090],{},[1003,2262,1093],{"stretchy":1086},[1010,2264,2041],{"encoding":1012},[662,2266,2268],{"className":2267,"ariaHidden":1018},[1017],[662,2269,2271,2274,2277,2280,2285,2288,2291],{"className":2270},[1022],[662,2272],{"className":2273,"style":1106},[1026],[662,2275,1083],{"className":2276,"style":1111},[1031,1110],[662,2278,1087],{"className":2279},[1115],[662,2281,1991,2283],{"className":2282},[1990],[662,2284,1995],{"style":1994},[662,2286],{"className":2287,"style":1999},[1723],[662,2289,1090],{"className":2290},[1031,1119],[662,2292,1093],{"className":2293},[1123]," s'il est équilibré.",[593,2296,2298],{"id":2297},"à-mémoriser","À mémoriser",[1138,2300,2301,2530,2766,2815,2876],{},[1141,2302,2303,2306,2307,2381,2382,609],{},[602,2304,2305],{},"Taille"," : ",[662,2308,2310,2337],{"className":2309},[987],[662,2311,2313],{"className":2312},[991],[993,2314,2315],{"xmlns":995},[997,2316,2317,2334],{},[1000,2318,2319,2322,2324,2328,2330,2332],{},[1080,2320,2321],{},"T",[1003,2323,1087],{"stretchy":1086},[2325,2326,2327],"mtext",{},"vide",[1003,2329,1093],{"stretchy":1086},[1003,2331,693],{},[1007,2333,1050],{},[1010,2335,2336],{"encoding":1012},"T(\\text{vide}) = 0",[662,2338,2340,2372],{"className":2339,"ariaHidden":1018},[1017],[662,2341,2343,2346,2350,2353,2360,2363,2366,2369],{"className":2342},[1022],[662,2344],{"className":2345,"style":1106},[1026],[662,2347,2321],{"className":2348,"style":2349},[1031,1119],"margin-right:0.1389em;",[662,2351,1087],{"className":2352},[1115],[662,2354,2357],{"className":2355},[1031,2356],"text",[662,2358,2327],{"className":2359},[1031],[662,2361,1093],{"className":2362},[1123],[662,2364],{"className":2365,"style":1724},[1723],[662,2367,693],{"className":2368},[1719],[662,2370],{"className":2371,"style":1724},[1723],[662,2373,2375,2378],{"className":2374},[1022],[662,2376],{"className":2377,"style":1062},[1026],[662,2379,1050],{"className":2380},[1031]," ; ",[662,2383,2385,2431],{"className":2384},[987],[662,2386,2388],{"className":2387},[991],[993,2389,2390],{"xmlns":995},[997,2391,2392,2428],{},[1000,2393,2394,2396,2398,2400,2402,2404,2406,2408,2410,2412,2415,2417,2419,2421,2423,2426],{},[1080,2395,2321],{},[1003,2397,1087],{"stretchy":1086},[1080,2399,1090],{},[1003,2401,1093],{"stretchy":1086},[1003,2403,693],{},[1007,2405,821],{},[1003,2407,923],{},[1080,2409,2321],{},[1003,2411,1087],{"stretchy":1086},[2325,2413,2414],{},"gauche",[1003,2416,1093],{"stretchy":1086},[1003,2418,923],{},[1080,2420,2321],{},[1003,2422,1087],{"stretchy":1086},[2325,2424,2425],{},"droite",[1003,2427,1093],{"stretchy":1086},[1010,2429,2430],{"encoding":1012},"T(n) = 1 + T(\\text{gauche}) + T(\\text{droite})",[662,2432,2434,2461,2479,2509],{"className":2433,"ariaHidden":1018},[1017],[662,2435,2437,2440,2443,2446,2449,2452,2455,2458],{"className":2436},[1022],[662,2438],{"className":2439,"style":1106},[1026],[662,2441,2321],{"className":2442,"style":2349},[1031,1119],[662,2444,1087],{"className":2445},[1115],[662,2447,1090],{"className":2448},[1031,1119],[662,2450,1093],{"className":2451},[1123],[662,2453],{"className":2454,"style":1724},[1723],[662,2456,693],{"className":2457},[1719],[662,2459],{"className":2460,"style":1724},[1723],[662,2462,2464,2467,2470,2473,2476],{"className":2463},[1022],[662,2465],{"className":2466,"style":1027},[1026],[662,2468,821],{"className":2469},[1031],[662,2471],{"className":2472,"style":2123},[1723],[662,2474,923],{"className":2475},[2127],[662,2477],{"className":2478,"style":2123},[1723],[662,2480,2482,2485,2488,2491,2497,2500,2503,2506],{"className":2481},[1022],[662,2483],{"className":2484,"style":1106},[1026],[662,2486,2321],{"className":2487,"style":2349},[1031,1119],[662,2489,1087],{"className":2490},[1115],[662,2492,2494],{"className":2493},[1031,2356],[662,2495,2414],{"className":2496},[1031],[662,2498,1093],{"className":2499},[1123],[662,2501],{"className":2502,"style":2123},[1723],[662,2504,923],{"className":2505},[2127],[662,2507],{"className":2508,"style":2123},[1723],[662,2510,2512,2515,2518,2521,2527],{"className":2511},[1022],[662,2513],{"className":2514,"style":1106},[1026],[662,2516,2321],{"className":2517,"style":2349},[1031,1119],[662,2519,1087],{"className":2520},[1115],[662,2522,2524],{"className":2523},[1031,2356],[662,2525,2425],{"className":2526},[1031],[662,2528,1093],{"className":2529},[1123],[1141,2531,2532,2306,2535,2381,2611,609],{},[602,2533,2534],{},"Hauteur",[662,2536,2538,2565],{"className":2537},[987],[662,2539,2541],{"className":2540},[991],[993,2542,2543],{"xmlns":995},[997,2544,2545,2562],{},[1000,2546,2547,2550,2552,2554,2556,2558,2560],{},[1080,2548,2549],{},"H",[1003,2551,1087],{"stretchy":1086},[2325,2553,2327],{},[1003,2555,1093],{"stretchy":1086},[1003,2557,693],{},[1003,2559,1005],{},[1007,2561,821],{},[1010,2563,2564],{"encoding":1012},"H(\\text{vide}) = -1",[662,2566,2568,2599],{"className":2567,"ariaHidden":1018},[1017],[662,2569,2571,2574,2578,2581,2587,2590,2593,2596],{"className":2570},[1022],[662,2572],{"className":2573,"style":1106},[1026],[662,2575,2549],{"className":2576,"style":2577},[1031,1119],"margin-right:0.0813em;",[662,2579,1087],{"className":2580},[1115],[662,2582,2584],{"className":2583},[1031,2356],[662,2585,2327],{"className":2586},[1031],[662,2588,1093],{"className":2589},[1123],[662,2591],{"className":2592,"style":1724},[1723],[662,2594,693],{"className":2595},[1719],[662,2597],{"className":2598,"style":1724},[1723],[662,2600,2602,2605,2608],{"className":2601},[1022],[662,2603],{"className":2604,"style":1027},[1026],[662,2606,1005],{"className":2607},[1031],[662,2609,821],{"className":2610},[1031],[662,2612,2614,2668],{"className":2613},[987],[662,2615,2617],{"className":2616},[991],[993,2618,2619],{"xmlns":995},[997,2620,2621,2665],{},[1000,2622,2623,2625,2627,2629,2631,2633,2635,2637,2640,2642,2644,2646,2648,2650,2652,2655,2657,2659,2661,2663],{},[1080,2624,2549],{},[1003,2626,1087],{"stretchy":1086},[1080,2628,1090],{},[1003,2630,1093],{"stretchy":1086},[1003,2632,693],{},[1007,2634,821],{},[1003,2636,923],{},[1080,2638,2639],{},"max",[1003,2641,1964],{},[1003,2643,1087],{"stretchy":1086},[1080,2645,2549],{},[1003,2647,1087],{"stretchy":1086},[2325,2649,2414],{},[1003,2651,1093],{"stretchy":1086},[1003,2653,2654],{"separator":1018},",",[1080,2656,2549],{},[1003,2658,1087],{"stretchy":1086},[2325,2660,2425],{},[1003,2662,1093],{"stretchy":1086},[1003,2664,1093],{"stretchy":1086},[1010,2666,2667],{"encoding":1012},"H(n) = 1 + \\max(H(\\text{gauche}), H(\\text{droite}))",[662,2669,2671,2698,2716],{"className":2670,"ariaHidden":1018},[1017],[662,2672,2674,2677,2680,2683,2686,2689,2692,2695],{"className":2673},[1022],[662,2675],{"className":2676,"style":1106},[1026],[662,2678,2549],{"className":2679,"style":2577},[1031,1119],[662,2681,1087],{"className":2682},[1115],[662,2684,1090],{"className":2685},[1031,1119],[662,2687,1093],{"className":2688},[1123],[662,2690],{"className":2691,"style":1724},[1723],[662,2693,693],{"className":2694},[1719],[662,2696],{"className":2697,"style":1724},[1723],[662,2699,2701,2704,2707,2710,2713],{"className":2700},[1022],[662,2702],{"className":2703,"style":1027},[1026],[662,2705,821],{"className":2706},[1031],[662,2708],{"className":2709,"style":2123},[1723],[662,2711,923],{"className":2712},[2127],[662,2714],{"className":2715,"style":2123},[1723],[662,2717,2719,2722,2725,2728,2731,2734,2740,2743,2747,2750,2753,2756,2762],{"className":2718},[1022],[662,2720],{"className":2721,"style":1106},[1026],[662,2723,2639],{"className":2724},[1990],[662,2726,1087],{"className":2727},[1115],[662,2729,2549],{"className":2730,"style":2577},[1031,1119],[662,2732,1087],{"className":2733},[1115],[662,2735,2737],{"className":2736},[1031,2356],[662,2738,2414],{"className":2739},[1031],[662,2741,1093],{"className":2742},[1123],[662,2744,2654],{"className":2745},[2746],"mpunct",[662,2748],{"className":2749,"style":1999},[1723],[662,2751,2549],{"className":2752,"style":2577},[1031,1119],[662,2754,1087],{"className":2755},[1115],[662,2757,2759],{"className":2758},[1031,2356],[662,2760,2425],{"className":2761},[1031],[662,2763,2765],{"className":2764},[1123],"))",[1141,2767,2768,2306,2771,2814],{},[602,2769,2770],{},"Coût des parcours",[662,2772,2774,2793],{"className":2773},[987],[662,2775,2777],{"className":2776},[991],[993,2778,2779],{"xmlns":995},[997,2780,2781,2791],{},[1000,2782,2783,2785,2787,2789],{},[1080,2784,1083],{"mathvariant":1082},[1003,2786,1087],{"stretchy":1086},[1080,2788,1090],{},[1003,2790,1093],{"stretchy":1086},[1010,2792,1096],{"encoding":1012},[662,2794,2796],{"className":2795,"ariaHidden":1018},[1017],[662,2797,2799,2802,2805,2808,2811],{"className":2798},[1022],[662,2800],{"className":2801,"style":1106},[1026],[662,2803,1083],{"className":2804,"style":1111},[1031,1110],[662,2806,1087],{"className":2807},[1115],[662,2809,1090],{"className":2810},[1031,1119],[662,2812,1093],{"className":2813},[1123],", chaque nœud visité une fois.",[1141,2816,2817,2820,2821,609],{},[602,2818,2819],{},"ABR équilibré"," : recherche et insertion en ",[662,2822,2824,2847],{"className":2823},[987],[662,2825,2827],{"className":2826},[991],[993,2828,2829],{"xmlns":995},[997,2830,2831,2845],{},[1000,2832,2833,2835,2837,2839,2841,2843],{},[1080,2834,1083],{"mathvariant":1082},[1003,2836,1087],{"stretchy":1086},[1080,2838,1961],{},[1003,2840,1964],{},[1080,2842,1090],{},[1003,2844,1093],{"stretchy":1086},[1010,2846,2041],{"encoding":1012},[662,2848,2850],{"className":2849,"ariaHidden":1018},[1017],[662,2851,2853,2856,2859,2862,2867,2870,2873],{"className":2852},[1022],[662,2854],{"className":2855,"style":1106},[1026],[662,2857,1083],{"className":2858,"style":1111},[1031,1110],[662,2860,1087],{"className":2861},[1115],[662,2863,1991,2865],{"className":2864},[1990],[662,2866,1995],{"style":1994},[662,2868],{"className":2869,"style":1999},[1723],[662,2871,1090],{"className":2872},[1031,1119],[662,2874,1093],{"className":2875},[1123],[1141,2877,2878,2881,2882,609],{},[602,2879,2880],{},"ABR dégénéré"," (clés déjà triées) : retombe à ",[662,2883,2885,2904],{"className":2884},[987],[662,2886,2888],{"className":2887},[991],[993,2889,2890],{"xmlns":995},[997,2891,2892,2902],{},[1000,2893,2894,2896,2898,2900],{},[1080,2895,1083],{"mathvariant":1082},[1003,2897,1087],{"stretchy":1086},[1080,2899,1090],{},[1003,2901,1093],{"stretchy":1086},[1010,2903,1096],{"encoding":1012},[662,2905,2907],{"className":2906,"ariaHidden":1018},[1017],[662,2908,2910,2913,2916,2919,2922],{"className":2909},[1022],[662,2911],{"className":2912,"style":1106},[1026],[662,2914,1083],{"className":2915,"style":1111},[1031,1110],[662,2917,1087],{"className":2918},[1115],[662,2920,1090],{"className":2921},[1031,1119],[662,2923,1093],{"className":2924},[1123],[593,2926,2928],{"id":2927},"les-quatre-parcours","Les quatre parcours",[1163,2930,2931,2942],{},[1166,2932,2933],{},[1169,2934,2935,2937,2939],{},[1172,2936,1174],{},[1172,2938,1177],{},[1172,2940,2941],{},"Sur ABR ?",[1182,2943,2944,2954,2966,2976],{},[1169,2945,2946,2948,2951],{},[1187,2947,1191],{},[1187,2949,2950],{},"racine, gauche, droite",[1187,2952,2953],{},"duplique l'arbre",[1169,2955,2956,2958,2961],{},[1187,2957,1204],{},[1187,2959,2960],{},"gauche, racine, droite",[1187,2962,2963],{},[602,2964,2965],{},"trie en ordre croissant",[1169,2967,2968,2970,2973],{},[1187,2969,1217],{},[1187,2971,2972],{},"gauche, droite, racine",[1187,2974,2975],{},"utile en évaluation d'expressions",[1169,2977,2978,2981,2984],{},[1187,2979,2980],{},"Largeur (BFS)",[1187,2982,2983],{},"niveau par niveau (file)",[1187,2985,2986],{},"aucun ordre particulier",[593,2988,2990],{"id":2989},"à-savoir-reconnaître","À savoir reconnaître",[1163,2992,2993,3003],{},[1166,2994,2995],{},[1169,2996,2997,3000],{},[1172,2998,2999],{},"Situation",[1172,3001,3002],{},"Action à mener",[1182,3004,3005,3013,3078,3089],{},[1169,3006,3007,3010],{},[1187,3008,3009],{},"Énumérer les valeurs triées d'un ABR",[1187,3011,3012],{},"parcours infixe",[1169,3014,3015,3018],{},[1187,3016,3017],{},"Tester si une clé est présente",[1187,3019,3020,3021,3049,3050],{},"descente récursive selon ",[662,3022,3024,3037],{"className":3023},[987],[662,3025,3027],{"className":3026},[991],[993,3028,3029],{"xmlns":995},[997,3030,3031,3035],{},[1000,3032,3033],{},[1003,3034,1700],{},[1010,3036,1700],{"encoding":1012},[662,3038,3040],{"className":3039,"ariaHidden":1018},[1017],[662,3041,3043,3046],{"className":3042},[1022],[662,3044],{"className":3045,"style":1715},[1026],[662,3047,1700],{"className":3048},[1719]," ou ",[662,3051,3053,3066],{"className":3052},[987],[662,3054,3056],{"className":3055},[991],[993,3057,3058],{"xmlns":995},[997,3059,3060,3064],{},[1000,3061,3062],{},[1003,3063,1749],{},[1010,3065,1749],{"encoding":1012},[662,3067,3069],{"className":3068,"ariaHidden":1018},[1017],[662,3070,3072,3075],{"className":3071},[1022],[662,3073],{"className":3074,"style":1715},[1026],[662,3076,1749],{"className":3077},[1719],[1169,3079,3080,3083],{},[1187,3081,3082],{},"Visiter niveau par niveau",[1187,3084,3085,3086],{},"BFS avec une ",[645,3087,3088],{},"deque",[1169,3090,3091,3094],{},[1187,3092,3093],{},"Calculer la profondeur maximale",[1187,3095,3096],{},"hauteur récursive",[593,3098,3100],{"id":3099},"pièges-courants","Pièges courants",[1138,3102,3103,3106,3112],{},[1141,3104,3105],{},"Confondre taille (nombre de nœuds) et hauteur (longueur du plus long chemin).",[1141,3107,3108,3109,609],{},"Oublier le cas de base ",[645,3110,3111],{},"if a is None: return",[1141,3113,3114,3115,3170],{},"Croire que l'ABR est toujours en ",[662,3116,3118,3141],{"className":3117},[987],[662,3119,3121],{"className":3120},[991],[993,3122,3123],{"xmlns":995},[997,3124,3125,3139],{},[1000,3126,3127,3129,3131,3133,3135,3137],{},[1080,3128,1083],{"mathvariant":1082},[1003,3130,1087],{"stretchy":1086},[1080,3132,1961],{},[1003,3134,1964],{},[1080,3136,1090],{},[1003,3138,1093],{"stretchy":1086},[1010,3140,2041],{"encoding":1012},[662,3142,3144],{"className":3143,"ariaHidden":1018},[1017],[662,3145,3147,3150,3153,3156,3161,3164,3167],{"className":3146},[1022],[662,3148],{"className":3149,"style":1106},[1026],[662,3151,1083],{"className":3152,"style":1111},[1031,1110],[662,3154,1087],{"className":3155},[1115],[662,3157,1991,3159],{"className":3158},[1990],[662,3160,1995],{"style":1994},[662,3162],{"className":3163,"style":1999},[1723],[662,3165,1090],{"className":3166},[1031,1119],[662,3168,1093],{"className":3169},[1123]," — c'est faux en cas\nde dégénérescence.",[593,3172,3174],{"id":3173},"syntaxe-python-à-retenir","Syntaxe Python (à retenir)",[653,3176,3178],{"className":655,"code":3177,"language":657,"meta":658,"style":658},"class Noeud:\n    def __init__(self, v, g=None, d=None):\n        self.valeur, self.gauche, self.droite = v, g, d\n\ndef infixe(a):\n    if a is None: return\n    infixe(a.gauche); print(a.valeur); infixe(a.droite)\n",[645,3179,3180,3188,3210,3232,3236,3244,3258],{"__ignoreMap":658},[662,3181,3182,3184,3186],{"class":664,"line":665},[662,3183,669],{"class":668},[662,3185,673],{"class":672},[662,3187,677],{"class":676},[662,3189,3190,3192,3194,3197,3199,3201,3204,3206,3208],{"class":664,"line":680},[662,3191,683],{"class":668},[662,3193,687],{"class":686},[662,3195,3196],{"class":676},"(self, v, g",[662,3198,693],{"class":668},[662,3200,651],{"class":686},[662,3202,3203],{"class":676},", d",[662,3205,693],{"class":668},[662,3207,651],{"class":686},[662,3209,705],{"class":676},[662,3211,3212,3214,3217,3220,3223,3225,3227,3229],{"class":664,"line":708},[662,3213,711],{"class":686},[662,3215,3216],{"class":676},".valeur, ",[662,3218,3219],{"class":686},"self",[662,3221,3222],{"class":676},".gauche, ",[662,3224,3219],{"class":686},[662,3226,740],{"class":676},[662,3228,693],{"class":668},[662,3230,3231],{"class":676}," v, g, d\n",[662,3233,3234],{"class":664,"line":722},[662,3235,752],{"emptyLinePlaceholder":751},[662,3237,3238,3240,3242],{"class":664,"line":735},[662,3239,876],{"class":668},[662,3241,1295],{"class":672},[662,3243,882],{"class":676},[662,3245,3246,3248,3250,3252,3254,3256],{"class":664,"line":748},[662,3247,887],{"class":668},[662,3249,890],{"class":676},[662,3251,893],{"class":668},[662,3253,896],{"class":686},[662,3255,1250],{"class":676},[662,3257,1253],{"class":668},[662,3259,3260,3263,3266],{"class":664,"line":755},[662,3261,3262],{"class":676},"    infixe(a.gauche); ",[662,3264,3265],{"class":686},"print",[662,3267,3268],{"class":676},"(a.valeur); infixe(a.droite)\n",[2197,3270,3271],{},"html pre.shiki code .s8jYJ, html code.shiki .s8jYJ{--shiki-light:#D73A49;--shiki-default:#D73A49;--shiki-dark:#F97583}html pre.shiki code .snPdu, html code.shiki .snPdu{--shiki-light:#6F42C1;--shiki-default:#6F42C1;--shiki-dark:#B392F0}html pre.shiki code .sxrX7, html code.shiki .sxrX7{--shiki-light:#24292E;--shiki-default:#24292E;--shiki-dark:#E1E4E8}html pre.shiki code .sBjJW, html code.shiki .sBjJW{--shiki-light:#005CC5;--shiki-default:#005CC5;--shiki-dark:#79B8FF}html .light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: var(--shiki-light-font-weight);text-decoration: var(--shiki-light-text-decoration);}html.light .shiki span {color: var(--shiki-light);background: var(--shiki-light-bg);font-style: var(--shiki-light-font-style);font-weight: 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g\n",[662,3390,3391,3393,3395,3397],{"class":664,"line":735},[662,3392,711],{"class":686},[662,3394,740],{"class":676},[662,3396,693],{"class":668},[662,3398,3399],{"class":676}," d\n",[3300,3401],{},[3293,3403,855],{"id":854},[1138,3405,3406,3411,3416,3419],{},[1141,3407,3408],{},[645,3409,3410],{},"taille(vide) = 0",[1141,3412,3413],{},[645,3414,3415],{},"hauteur(vide) = -1",[1141,3417,3418],{},"Récursion sur les deux sous-arbres.",[1141,3420,3421,3422,609],{},"Coût : 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(1ᵉ)\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-te-algorithmique-abr-infixe-flowchart-N3-4\" data-look=\"classic\" transform=\"translate(227.40625, 252.5)\">\u003Crect class=\"basic label-container\" style=\"\" x=\"-58.640625\" y=\"-22.5\" width=\"117.28125\" height=\"45\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-26.640625, -10.5)\">\u003Crect\u002F>\u003CforeignObject width=\"53.28125\" height=\"21\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table-cell; white-space: nowrap; line-height: 1.5; max-width: 200px; text-align: center;\">\u003Cspan class=\"nodeLabel\">\u003Cp>3 — (3ᵉ)\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003Cg class=\"node default\" id=\"ill-2-nsi-tale-te-algorithmique-abr-infixe-flowchart-N7-5\" data-look=\"classic\" transform=\"translate(392.35546875, 252.5)\">\u003Crect class=\"basic label-container\" style=\"\" x=\"-58.296875\" y=\"-22.5\" width=\"116.59375\" height=\"45\"\u002F>\u003Cg class=\"label\" style=\"\" transform=\"translate(-26.296875, -10.5)\">\u003Crect\u002F>\u003CforeignObject width=\"52.59375\" height=\"21\">\u003Cdiv xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1999\u002Fxhtml\" style=\"display: table-cell; white-space: nowrap; line-height: 1.5; max-width: 200px; text-align: center;\">\u003Cspan class=\"nodeLabel\">\u003Cp>7 — (6ᵉ)\u003C\u002Fp>\u003C\u002Fspan>\u003C\u002Fdiv>\u003C\u002FforeignObject>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003C\u002Fg>\u003Cdefs>\u003Cfilter id=\"ill-2-nsi-tale-te-algorithmique-abr-infixe-drop-shadow\" height=\"130%\" width=\"130%\">\u003CfeDropShadow dx=\"4\" dy=\"4\" stdDeviation=\"0\" flood-opacity=\"0.06\" flood-color=\"#000000\"\u002F>\u003C\u002Ffilter>\u003C\u002Fdefs>\u003Cdefs>\u003Cfilter id=\"ill-2-nsi-tale-te-algorithmique-abr-infixe-drop-shadow-small\" height=\"150%\" width=\"150%\">\u003CfeDropShadow dx=\"2\" dy=\"2\" stdDeviation=\"0\" flood-opacity=\"0.06\" flood-color=\"#000000\"\u002F>\u003C\u002Ffilter>\u003C\u002Fdefs>\u003ClinearGradient id=\"ill-2-nsi-tale-te-algorithmique-abr-infixe-gradient\" gradientUnits=\"objectBoundingBox\" x1=\"0%\" y1=\"0%\" x2=\"100%\" y2=\"0%\">\u003Cstop offset=\"0%\" stop-color=\"#4948b8\" stop-opacity=\"1\"\u002F>\u003Cstop offset=\"100%\" stop-color=\"#c4c8cd\" stop-opacity=\"1\"\u002F>\u003C\u002FlinearGradient>\u003C\u002Fsvg>",[598,3513,3514,3515,609],{},"L'infixe d'un ABR ",[602,3516,3517],{},"trie",[3300,3519],{},[3293,3521,1406],{"id":1405},[1138,3523,3524,3533,3536,3539],{},[1141,3525,3526,3527,3529,3530,609],{},"Une ",[602,3528,1421],{}," (FIFO) — ",[645,3531,3532],{},"collections.deque",[1141,3534,3535],{},"À chaque tour : on défile, on enfile les enfants.",[1141,3537,3538],{},"Visite niveau par niveau.",[1141,3540,3421,3541,609],{},[662,3542,3544,3563],{"className":3543},[987],[662,3545,3547],{"className":3546},[991],[993,3548,3549],{"xmlns":995},[997,3550,3551,3561],{},[1000,3552,3553,3555,3557,3559],{},[1080,3554,1083],{"mathvariant":1082},[1003,3556,1087],{"stretchy":1086},[1080,3558,1090],{},[1003,3560,1093],{"stretchy":1086},[1010,3562,1096],{"encoding":1012},[662,3564,3566],{"className":3565,"ariaHidden":1018},[1017],[662,3567,3569,3572,3575,3578,3581],{"className":3568},[1022],[662,3570],{"className":3571,"style":1106},[1026],[662,3573,1083],{"className":3574,"style":1111},[1031,1110],[662,3576,1087],{"className":3577},[1115],[662,3579,1090],{"className":3580},[1031,1119],[662,3582,1093],{"className":3583},[1123],[3300,3585],{},[3293,3587,3589],{"id":3588},"abr-rechercher","ABR : rechercher",[598,3591,3592,3593,3595],{},"À chaque comparaison, on ",[602,3594,1935],{}," des descendants.",[1138,3597,3598,3656],{},[1141,3599,3600,3601,609],{},"Équilibré → 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",[662,3660,3662,3681],{"className":3661},[987],[662,3663,3665],{"className":3664},[991],[993,3666,3667],{"xmlns":995},[997,3668,3669,3679],{},[1000,3670,3671,3673,3675,3677],{},[1080,3672,1083],{"mathvariant":1082},[1003,3674,1087],{"stretchy":1086},[1080,3676,1090],{},[1003,3678,1093],{"stretchy":1086},[1010,3680,1096],{"encoding":1012},[662,3682,3684],{"className":3683,"ariaHidden":1018},[1017],[662,3685,3687,3690,3693,3696,3699],{"className":3686},[1022],[662,3688],{"className":3689,"style":1106},[1026],[662,3691,1083],{"className":3692,"style":1111},[1031,1110],[662,3694,1087],{"className":3695},[1115],[662,3697,1090],{"className":3698},[1031,1119],[662,3700,1093],{"className":3701},[1123],[3300,3703],{},[3293,3705,3707],{"id":3706},"abr-insérer","ABR : insérer",[653,3709,3711],{"className":655,"code":3710,"language":657,"meta":658,"style":658},"def inserer(a, cle):\n    if a is None: return Noeud(cle)\n    if cle \u003C a.valeur:  a.gauche = inserer(a.gauche, cle)\n    elif cle > 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