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Perfect Binary Tree

Perfect Binary Tree

In this tutorial, you will learn about the perfect binary tree. Also, you will find working examples for checking a perfect binary tree in C, C++, Java and Python.

A perfect binary tree is a type of binary tree in which every internal node has exactly two child nodes and all the leaf nodes are at the same level.

Perfect Binary Tree
Perfect Binary Tree

All the internal nodes have a degree of 2.

Recursively, a perfect binary tree can be defined as:

  1. If a single node has no children, it is a perfect binary tree of height h = 0,
  2. If a node has h > 0, it is a perfect binary tree if both of its subtrees are of height h - 1 and are non-overlapping.
Perfect Binary Tree (Recursive Representation)
Perfect Binary Tree (Recursive Representation)

Python, Java and C/C++ Examples

The following code is for checking whether a tree is a perfect binary tree.

# 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 <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 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\n");
  else
    printf("The tree is not a perfect binary tree\n");
}
// Checking 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 a perfect binary tree\n";
  else
    cout << "The tree is not a perfect binary tree\n";
}

Perfect Binary Tree Theorems

  1. A perfect binary tree of height h has 2h + 1 – 1 node.
  2. A perfect binary tree with n nodes has height log(n + 1) – 1 = Θ(ln(n)).
  3. A perfect binary tree of height h has 2h leaf nodes.
  4. The average depth of a node in a perfect binary tree is Θ(ln(n)).