Algorithm in C – Perfect Binary Tree

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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.

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)

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 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 treen");
  else
    printf("The tree is not a perfect binary treen");
}

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)).

 

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