Autour des arbres binaires (2)

Différentes représentations

Cet exercice demande d'écrire les méthodes calculant la taille et hauteur d'un arbre binaire. Il fait partie d'une série dans laquelle on utilise différentes représentations :

On rappelle qu'un arbre binaire est :

soit vide ;
soit composé d'un nœud appelé racine et ayant une valeur, et de deux arbres binaires appelés sous-arbre gauche et sous-arbre droit de la racine.

On représente dans cet exercice les arbres binaires à l'aide d'une classe AB ne possédant qu'un seul attribut racine.

Si None est affecté à cet attribut, alors l'arbre binaire est vide.

Dans le cas où l'arbre binaire n'est pas vide, son attribut racine est un objet de type Noeudqui a trois attributs :

valeur qui est la valeur portée par la racine ;
gauche est un objet de type AB et représente le sous-arbre gauche de l'arbre ;
droit est un objet de type AB et représente le sous-arbre droit de l'arbre.

Les classes Noeud et AB

class Noeud:
    def __init__(self, gauche, valeur, droit):
        self.gauche = gauche
        self.valeur = valeur
        self.droit = droit


class AB:
    def __init__(self, gauche, racine, droit):
        if racine is not None:
            if gauche is None:
                gauche = AB(None, None, None)
            if droit is None:
                droit = AB(None, None, None)
            racine = Noeud(gauche, racine, droit)
        self.racine = racine

    def est_vide(self):
        return self.racine is None

    def __str__(self):
        if self.est_vide():
            return "∅"
        else:
            return f"({self.racine.gauche}, {self.racine.valeur}, {self.racine.droit})"

Créer un arbre binaire

Les deux classes Noeud et AB décrites ci-dessus est déjà importées dans l'éditeur.

Utiliser ces classes afin de représenter les deux arbres binaires ci-dessous :

flowchart TD
    A0((0))
    B0((1))
    C0((2))
    D0((3))
    E0[ ]
    F0((4))
    G0((5))

    A0 --> B0
    A0 --> C0
    B0 --> D0
    B0 ~~~ E0
    C0 --> F0
    C0 --> G0

    a((a))
    b((b))
    c((c))
    d(( ))
    e((e))
    f((f))
    g(( ))
    h((h))
    i(( ))
    j(( ))
    k((k))

    a --> b
    a --> c
    b ~~~ d
    b --> e
    c --> f
    c ~~~ g
    e --> h
    e ~~~ i
    f ~~~ j
    f --> k

    style E0 fill:none, stroke:none
    style d fill:none, stroke:none
    style g fill:none, stroke:none
    style i fill:none, stroke:none
    style j fill:none, stroke:none

Par commodité, on n'a pas représenté les arbres vides.

ab_int sera l'arbre portant des nombres entiers, ab_str celui portant des caractères.

Évaluations restantes : 10/10

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Taille

Compléter la méthode taille de la classe AB. Cette méthode renvoie la taille de l'arbre binaire représenté par l'objet en question.

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.taille()
0
>>> ab.taille()
3

Évaluations restantes : 10/10

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Hauteur

Compléter la méthode hauteur de la classe AB. Cette méthode renvoie la hauteur de l'arbre binaire représenté par l'objet en question.

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.hauteur()
0
>>> ab.hauteur()
3

Évaluations restantes : 10/10

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Parcours en largeur

Compléter la méthode largeur de la classe AB. Cette méthode renvoie la liste des valeurs des nœuds rencontrés lors du parcours en largeur de l'arbre binaire représenté par l'objet en question.

La classe File décrite ci-dessous est déjà importée dans l'éditeur.

Interface de la classe File

f = File() : crée une file vide et l'affecte à la variable f;
f.est_vide() : renvoie le booléen indiquant si la file f est vide ;
f.enfile(x) : enfile l'élément x dans la file f;
f.defile() : défile un élément de la file f si elle n'est pas vide et le renvoie. Provoque une erreur si la file est vide.

La classe File

from collections import deque


class File:
    """Classe définissant une structure de file"""

    def __init__(self):
        self.valeurs = deque([])

    def est_vide(self):
        """Renvoie le booléen True si la file est vide, False sinon"""
        return len(self.valeurs) == 0

    def enfile(self, x):
        """Place x à la queue de la file"""
        self.valeurs.appendleft(x)

    def defile(self):
        """Retire et renvoie l'élément placé à la tête de la file.
        Provoque une erreur si la file est vide
        """
        if self.est_vide():
            raise ValueError("La file est vide")
        return self.valeurs.pop()

    def __repr__(self):
        """Convertit la file en une chaîne de caractères"""
        if self.est_vide():
            return "∅"
        return f"(queue) {' -> '.join(str(x) for x in self.valeurs)} (tête)"

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.largeur()
[]
>>> ab.largeur()
[0, 2, 1]

Évaluations restantes : 10/10

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Parcours en profondeur préfixe

Compléter la méthode prefixe de la classe AB. Cette méthode renvoie la liste des valeurs des nœuds rencontrés lors du parcours en profondeur préfixe de l'arbre binaire représenté par l'objet en question.

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.prefixe()
[]
>>> ab.prefixe()
[0, 2, 1]

Évaluations restantes : 10/10

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Parcours en profondeur infixe

Compléter la méthode infixe de la classe AB. Cette méthode renvoie la liste des valeurs des nœuds rencontrés lors du parcours en profondeur infixe de l'arbre binaire représenté par l'objet en question.

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.infixe()
[]
>>> ab.infixe()
[0, 1, 2]

Évaluations restantes : 10/10

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Parcours en profondeur suffixe

Compléter la méthode suffixe de la classe AB. Cette méthode renvoie la liste des valeurs des nœuds rencontrés lors du parcours en profondeur suffixe de l'arbre binaire représenté par l'objet en question.

Exemples

>>> vide = AB(None, None, None)
>>> ab_1 = AB(None, 1, None)
>>> ab = AB(None, 0, AB(ab_1, 2, None))
>>> vide.suffixe()
[]
>>> ab.suffixe()
[1, 2, 0]

Évaluations restantes : 10/10

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Compléments

Il est à noter qu'ici, il n'est pas possible de stocker None en tant que valeur dans un arbre binaire implémenté de cette façon. En effet, si l'argument valeur est None, l'instance interne de Noeud n'est jamais créée. Ce n'est généralement pas gênant, car il rare d'avoir besoin de stocker None en plus d'autres types de données dans le même arbre binaire, mais il peut être intéressant de garder ce détail à l'esprit.

Il est également possible de déclarer la classe Noeud à l'intérieur de la classe AB, de manière à ne pas l'exposer à l'utilisateur. Elle est alors un attribut de classe, qui peut être accédé depuis une instance avec self.Noeud (Nota: le comportement des attributs de classes vs instances a un fonctionnement particulier à python).
Dans ce cas, l'implantation de la méthode __init__ devient :

class AB:

    class Noeud:
        def __init__(self, gauche, valeur, droit):
            self.valeur = valeur
            self.gauche = None
            self.droit = None

    def __init__(self, gauche, racine, droit):
        if racine is not None:
            if gauche is None:
                gauche = AB(None, None, None)
            if droit is None:
                droit = AB(None, None, None)
            racine = self.Noeud(gauche, racine, droit)
        self.racine = racine