Perfect Binary Tree
Perfect Binary Tree
Perfect Binary Tree is defined as the binary tree in which all the internal nodes have two children and all leaf nodes are at the same level.
A tree is a PBT if all the internal nodes have 2 child, and all the leaf nodes are at the same level.
PBT Theorems
- A PBT of height h has
2
h + 1– 1
node. - A PBT with n nodes has height
log(n + 1) – 1 = Θ(ln(n))
. - A PBT of height h has
2
h leaf nodes. - The average depth of a node in a PBT is
Θ(ln(n))
Program of PBT in C
// To Check if a binary tree is a perfect binary tree in C #include <stdbool.h> #include <stdio.h> #include <stdlib.h> 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 whether 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); } 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 perfect binary tree\n"); else printf("The tree is not perfect binary tree\n"); }
Output
The tree is perfect binary tree
Program of PBT in C++
//To Check if a binary tree is a perfect binary tree in C++ #include <iostream> 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 perfect binary tree\n"; else cout << "The tree is not perfect binary tree\n"; }
Output
The tree is perfect binary tree
Program of PBT 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 whether the tree is perfect binary tree def is_perfect(root, d, level=0): # Check whether the tree is empty if (root is None): return True 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 perfect binary tree") else: print("The tree is not perfect binary tree")
Output
THe tree is perfect binary tree
Program of PBT 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 whether the tree is perfect binary tree static boolean is_perfect(Node root, int d, int level) { // Check whether 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); } 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 perfect binary tree"); else System.out.println("The tree is not perfect binary tree"); } }
Output
The tree is perfect binary tree
Read more, Full Binary Tree