Pohon Biner Sempurna

Dalam tutorial ini, Anda akan belajar tentang pohon biner yang sempurna. Selain itu, Anda akan menemukan contoh kerja untuk memeriksa pohon biner yang sempurna di C, C ++, Java dan Python.

Pohon biner sempurna adalah jenis pohon biner di mana setiap simpul internal memiliki tepat dua simpul anak dan semua simpul daun berada pada level yang sama.

Pohon Biner Sempurna

Semua node internal memiliki derajat 2.

Secara rekursif, pohon biner sempurna dapat didefinisikan sebagai:

  1. Jika satu node tidak memiliki anak, itu adalah pohon biner yang sempurna h = 0,
  2. Jika simpul memilikinya h> 0, itu adalah pohon biner yang sempurna jika kedua subpohonnya tinggi h - 1dan tidak tumpang tindih.
Pohon Biner Sempurna (Representasi Rekursif)

Contoh Python, Java dan C / C ++

Kode berikut untuk memeriksa apakah pohon adalah pohon biner yang sempurna.

Python Java C ++ 
 # Checking if a binary tree is a perfect binary tree in Python class newNode: def __init__(self, k): self.key = k self.right = self.left = None # Calculate the depth def calculateDepth(node): d = 0 while (node is not None): d += 1 node = node.left return d # Check if the tree is perfect binary tree def is_perfect(root, d, level=0): # Check if the tree is empty if (root is None): return True # Check the presence of trees if (root.left is None and root.right is None): return (d == level + 1) if (root.left is None or root.right is None): return False return (is_perfect(root.left, d, level + 1) and is_perfect(root.right, d, level + 1)) root = None root = newNode(1) root.left = newNode(2) root.right = newNode(3) root.left.left = newNode(4) root.left.right = newNode(5) if (is_perfect(root, calculateDepth(root))): print("The tree is a perfect binary tree") else: print("The tree is not a perfect binary tree")
 // Checking if a binary tree is a perfect binary tree in Java class PerfectBinaryTree ( static class Node ( int key; Node left, right; ) // Calculate the depth static int depth(Node node) ( int d = 0; while (node != null) ( d++; node = node.left; ) return d; ) // Check if the tree is perfect binary tree static boolean is_perfect(Node root, int d, int level) ( // Check if the tree is empty if (root == null) return true; // If for children if (root.left == null && root.right == null) return (d == level + 1); if (root.left == null || root.right == null) return false; return is_perfect(root.left, d, level + 1) && is_perfect(root.right, d, level + 1); ) // Wrapper function static boolean is_Perfect(Node root) ( int d = depth(root); return is_perfect(root, d, 0); ) // Create a new node static Node newNode(int k) ( Node node = new Node(); node.key = k; node.right = null; node.left = null; return node; ) public static void main(String args()) ( Node root = null; root = newNode(1); root.left = newNode(2); root.right = newNode(3); root.left.left = newNode(4); root.left.right = newNode(5); if (is_Perfect(root) == true) System.out.println("The tree is a perfect binary tree"); else System.out.println("The tree is not a perfect binary tree"); ) )
 // Checking if a binary tree is a perfect binary tree in C #include #include #include struct node ( int data; struct node *left; struct node *right; ); // Creating a new node struct node *newnode(int data) ( struct node *node = (struct node *)malloc(sizeof(struct node)); node->data = data; node->left = NULL; node->right = NULL; return (node); ) // Calculate the depth int depth(struct node *node) ( int d = 0; while (node != NULL) ( d++; node = node->left; ) return d; ) // Check if the tree is perfect bool is_perfect(struct node *root, int d, int level) ( // Check if the tree is empty if (root == NULL) return true; // Check the presence of children if (root->left == NULL && root->right == NULL) return (d == level + 1); if (root->left == NULL || root->right == NULL) return false; return is_perfect(root->left, d, level + 1) && is_perfect(root->right, d, level + 1); ) // Wrapper function bool is_Perfect(struct node *root) ( int d = depth(root); return is_perfect(root, d, 0); ) int main() ( struct node *root = NULL; root = newnode(1); root->left = newnode(2); root->right = newnode(3); root->left->left = newnode(4); root->left->right = newnode(5); root->right->left = newnode(6); if (is_Perfect(root)) printf("The tree is a perfect binary tree"); else printf("The tree is not a perfect binary tree"); )
 // Checking if a binary tree is a perfect binary tree in C++ #include using namespace std; struct Node ( int key; struct Node *left, *right; ); int depth(Node *node) ( int d = 0; while (node != NULL) ( d++; node = node->left; ) return d; ) bool isPerfectR(struct Node *root, int d, int level = 0) ( if (root == NULL) return true; if (root->left == NULL && root->right == NULL) return (d == level + 1); if (root->left == NULL || root->right == NULL) return false; return isPerfectR(root->left, d, level + 1) && isPerfectR(root->right, d, level + 1); ) bool isPerfect(Node *root) ( int d = depth(root); return isPerfectR(root, d); ) struct Node *newNode(int k) ( struct Node *node = new Node; node->key = k; node->right = node->left = NULL; return node; ) int main() ( struct Node *root = NULL; root = newNode(1); root->left = newNode(2); root->right = newNode(3); root->left->left = newNode(4); root->left->right = newNode(5); root->right->left = newNode(6); if (isPerfect(root)) cout << "The tree is a perfect binary tree"; else cout << "The tree is not a perfect binary tree"; )

Teorema Pohon Biner Sempurna

  1. Pohon biner sempurna dengan tinggi h memiliki simpul.2h + 1 - 1
  2. Sebuah pohon biner sempurna dengan n simpul memiliki tinggi log(n + 1) - 1 = Θ(ln(n)).
  3. Pohon biner sempurna dengan tinggi h memiliki simpul daun.2h
  4. Kedalaman rata-rata sebuah simpul dalam pohon biner yang sempurna adalah Θ(ln(n)).

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