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Check if a given Binary Tree is Heap

  • Difficulty Level : Medium
  • Last Updated : 02 Jul, 2021

Given a binary tree, we need to check it has heap property or not, Binary tree need to fulfill the following two conditions for being a heap – 

  1. It should be a complete tree (i.e. all levels except last should be full).
  2. Every node’s value should be greater than or equal to its child node (considering max-heap).

For example this tree contains heap property –
 

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yes



While this doesn’t –

no

We check each of the above condition separately, for checking completeness isComplete and for checking heap isHeapUtil function are written. 
Detail about isComplete function can be found here.
isHeapUtil function is written considering the following things –  

  1. Every Node can have 2 children, 0 child (last level nodes) or 1 child (there can be at most one such node).
  2. If Node has No child then it’s a leaf node and returns true (Base case)
  3. If Node has one child (it must be left child because it is a complete tree) then we need to compare this node with its single child only.
  4. If the Node has both child then check heap property at Node at recur for both subtrees. 
    Complete code.

Below is the implementation of the above approach:

C++




/* C++ program to checks if a
binary tree is max heap or not */
#include <bits/stdc++.h>
 
using namespace std;
  
/*  Tree node structure */
struct Node
{
    int key;
    struct Node *left;
    struct Node *right;
};
  
/* Helper function that
allocates a new node */
struct Node *newNode(int k)
{
    struct Node *node = new Node;
    node->key = k;
    node->right = node->left = NULL;
    return node;
}
  
/* This function counts the
number of nodes in a binary tree */
unsigned int countNodes(struct Node* root)
{
    if (root == NULL)
        return (0);
    return (1 + countNodes(root->left)
            + countNodes(root->right));
}
  
/* This function checks if the
binary tree is complete or not */
bool isCompleteUtil (struct Node* root,
                     unsigned int index,
                     unsigned int number_nodes)
{
    // An empty tree is complete
    if (root == NULL)
        return (true);
  
    // If index assigned to
    // current node is more than
    // number of nodes in tree,
    // then tree is not complete
    if (index >= number_nodes)
        return (false);
  
    // Recur for left and right subtrees
    return (isCompleteUtil(root->left, 2*index + 1,
                           number_nodes) &&
            isCompleteUtil(root->right, 2*index + 2,
                           number_nodes));
}
  
// This Function checks the
// heap property in the tree.
bool isHeapUtil(struct Node* root)
{
    //  Base case : single
    // node satisfies property
    if (root->left == NULL && root->right == NULL)
        return (true);
  
    //  node will be in
    // second last level
    if (root->right == NULL)
    {
        //  check heap property at Node
        //  No recursive call ,
        // because no need to check last level
        return (root->key >= root->left->key);
    }
    else
    {
        //  Check heap property at Node and
        //  Recursive check heap
        // property at left and right subtree
        if (root->key >= root->left->key &&
            root->key >= root->right->key)
            return ((isHeapUtil(root->left)) &&
                    (isHeapUtil(root->right)));
        else
            return (false);
    }
}
  
//  Function to check binary
// tree is a Heap or Not.
bool isHeap(struct Node* root)
{
    // These two are used
    // in isCompleteUtil()
    unsigned int node_count = countNodes(root);
    unsigned int index = 0;
  
    if (isCompleteUtil(root, index,
                       node_count)
        && isHeapUtil(root))
        return true;
    return false;
}
  
// Driver code
int main()
{
    struct Node* root = NULL;
    root = newNode(10);
    root->left = newNode(9);
    root->right = newNode(8);
    root->left->left = newNode(7);
    root->left->right = newNode(6);
    root->right->left = newNode(5);
    root->right->right = newNode(4);
    root->left->left->left = newNode(3);
    root->left->left->right = newNode(2);
    root->left->right->left = newNode(1);
  
    // Function call
    if (isHeap(root))
        cout << "Given binary tree is a Heap\n";
    else
        cout << "Given binary tree is not a Heap\n";
  
    return 0;
}
 
// This code is contributed by shubhamsingh10

C




/* C program to checks if a binary
   tree is max heap or not
 */
#include <stdbool.h>
#include <stdio.h>
#include <stdlib.h>
 
/*  Tree node structure */
struct Node {
    int key;
    struct Node* left;
    struct Node* right;
};
 
/* Helper function
that allocates a new node */
struct Node* newNode(int k)
{
    struct Node* node
        = (struct Node*)malloc(sizeof(struct Node));
    node->key = k;
    node->right = node->left = NULL;
    return node;
}
 
/* This function counts the number
   of nodes in a binary tree
 */
unsigned int countNodes(struct Node* root)
{
    if (root == NULL)
        return (0);
    return (1 + countNodes(root->left)
            + countNodes(root->right));
}
 
/* This function checks
   if the binary tree is complete or
 * not */
bool isCompleteUtil(struct Node* root,
                    unsigned int index,
                    unsigned int number_nodes)
{
    // An empty tree is complete
    if (root == NULL)
        return (true);
 
    // If index assigned to current
    // node is more than
    // number of nodes in tree,
    // then tree is not complete
    if (index >= number_nodes)
        return (false);
 
    // Recur for left and right subtrees
    return (isCompleteUtil(root->left,
                           2 * index + 1,
                           number_nodes)
            && isCompleteUtil(root->right,
                              2 * index + 2,
                              number_nodes));
}
 
// This Function checks the
// heap property in the tree.
bool isHeapUtil(struct Node* root)
{
    //  Base case : single
    // node satisfies property
    if (root->left == NULL && root->right == NULL)
        return (true);
 
    //  node will be in second last level
    if (root->right == NULL) {
        //  check heap property at Node
        //  No recursive call ,
        //  because no need to check last level
        return (root->key >= root->left->key);
    }
    else {
        //  Check heap property at Node and
        //  Recursive check heap property
        //   at left and right subtree
        if (root->key >= root->left->key
            && root->key >= root->right->key)
            return ((isHeapUtil(root->left))
                    && (isHeapUtil(root->right)));
        else
            return (false);
    }
}
 
//  Function to check binary
// tree is a Heap or Not.
bool isHeap(struct Node* root)
{
    // These two are used in
    // isCompleteUtil()
    unsigned int node_count = countNodes(root);
    unsigned int index = 0;
 
    if (isCompleteUtil(root, index, node_count)
        && isHeapUtil(root))
        return true;
    return false;
}
 
// Driver Code
int main()
{
    struct Node* root = NULL;
    root = newNode(10);
    root->left = newNode(9);
    root->right = newNode(8);
    root->left->left = newNode(7);
    root->left->right = newNode(6);
    root->right->left = newNode(5);
    root->right->right = newNode(4);
    root->left->left->left = newNode(3);
    root->left->left->right = newNode(2);
    root->left->right->left = newNode(1);
 
    if (isHeap(root))
        printf("Given binary tree is a Heap\n");
    else
        printf("Given binary tree is not a Heap\n");
 
    return 0;
}

Java




/* Java program to checks
 * if a binary tree is max heap or not */
 
// A Binary Tree node
class Node {
    int key;
    Node left, right;
 
    Node(int k)
    {
        key = k;
        left = right = null;
    }
}
 
class Is_BinaryTree_MaxHeap
{
    /* This function counts
       the number of nodes in a binary
     * tree */
    int countNodes(Node root)
    {
        if (root == null)
            return 0;
        return (1 + countNodes(root.left)
                + countNodes(root.right));
    }
 
    /* This function checks
       if the binary tree is complete
     * or not */
    boolean isCompleteUtil(Node root, int index,
                           int number_nodes)
    {
        // An empty tree is complete
        if (root == null)
            return true;
 
        // If index assigned to current
        //  node is more than number of
        //  nodes in tree,  then tree is
        // not complete
        if (index >= number_nodes)
            return false;
 
        // Recur for left and right subtrees
        return isCompleteUtil(root.left,
                              2 * index + 1,
                              number_nodes)
            && isCompleteUtil(root.right,
                              2 * index + 2,
                              number_nodes);
    }
 
    // This Function checks
    // the heap property in the tree.
    boolean isHeapUtil(Node root)
    {
        //  Base case : single
        // node satisfies property
        if (root.left == null && root.right == null)
            return true;
 
        //  node will be in second last level
        if (root.right == null) {
            //  check heap property at Node
            //  No recursive call ,
            //  because no need to check last level
            return root.key >= root.left.key;
        }
        else {
            //  Check heap property at Node and
            //  Recursive check heap property at left and
            //  right subtree
            if (root.key >= root.left.key
                && root.key >= root.right.key)
                return isHeapUtil(root.left)
                    && isHeapUtil(root.right);
            else
                return false;
        }
    }
 
    //  Function to check binary
    // tree is a Heap or Not.
    boolean isHeap(Node root)
    {
        if (root == null)
            return true;
 
        // These two are used
        // in isCompleteUtil()
        int node_count = countNodes(root);
 
        if (isCompleteUtil(root, 0, node_count) == true
            && isHeapUtil(root) == true)
            return true;
        return false;
    }
 
    // driver function to
    // test the above functions
    public static void main(String args[])
    {
        Is_BinaryTree_MaxHeap bt
            = new Is_BinaryTree_MaxHeap();
 
        Node root = new Node(10);
        root.left = new Node(9);
        root.right = new Node(8);
        root.left.left = new Node(7);
        root.left.right = new Node(6);
        root.right.left = new Node(5);
        root.right.right = new Node(4);
        root.left.left.left = new Node(3);
        root.left.left.right = new Node(2);
        root.left.right.left = new Node(1);
 
        if (bt.isHeap(root) == true)
            System.out.println(
                "Given binary tree is a Heap");
        else
            System.out.println(
                "Given binary tree is not a Heap");
    }
}
 
// This code has been contributed by Amit Khandelwal

Python




# To check if a binary tree
# is a MAX Heap or not
 
 
class GFG:
    def __init__(self, value):
        self.key = value
        self.left = None
        self.right = None
 
    def count_nodes(self, root):
        if root is None:
            return 0
        else:
            return (1 + self.count_nodes(root.left) +
                    self.count_nodes(root.right))
 
    def heap_propert_util(self, root):
 
        if (root.left is None and
                root.right is None):
            return True
 
        if root.right is None:
            return root.key >= root.left.key
        else:
            if (root.key >= root.left.key and
                    root.key >= root.right.key):
                return (self.heap_propert_util(root.left) and
                        self.heap_propert_util(root.right))
            else:
                return False
 
    def complete_tree_util(self, root,
                           index, node_count):
        if root is None:
            return True
        if index >= node_count:
            return False
        return (self.complete_tree_util(root.left, 2 *
                                        index + 1, node_count) and
                self.complete_tree_util(root.right, 2 *
                                        index + 2, node_count))
 
    def check_if_heap(self):
        node_count = self.count_nodes(self)
        if (self.complete_tree_util(self, 0, node_count) and
                self.heap_propert_util(self)):
            return True
        else:
            return False
 
 
# Driver Code
root = GFG(5)
root.left = GFG(2)
root.right = GFG(3)
root.left.left = GFG(1)
 
if root.check_if_heap():
    print("Given binary tree is a heap")
else:
    print("Given binary tree is not a Heap")
 
# This code has been
# contributed by Yash Agrawal

C#




/* C# program to checks if a
binary tree is max heap or not
 */
using System;
 
// A Binary Tree node
public class Node {
    public int key;
    public Node left, right;
 
    public Node(int k)
    {
        key = k;
        left = right = null;
    }
}
 
class Is_BinaryTree_MaxHeap
{
    /* This function counts the number
    of nodes in a binary tree */
    int countNodes(Node root)
    {
        if (root == null)
            return 0;
        return (1 + countNodes(root.left)
                + countNodes(root.right));
    }
 
    /* This function checks if the
    binary tree is complete or not */
    Boolean isCompleteUtil(Node root, int index,
                           int number_nodes)
    {
        // An empty tree is complete
        if (root == null)
            return true;
 
        // If index assigned to
        // current node is more than
        // number of nodes in tree, then
        // tree is notcomplete
        if (index >= number_nodes)
            return false;
 
        // Recur for left and right subtrees
        return isCompleteUtil(root.left,
                              2 * index + 1,
                              number_nodes)
            && isCompleteUtil(root.right,
                              2 * index + 2,
                              number_nodes);
    }
 
    // This Function checks the
    // heap property in the tree.
    Boolean isHeapUtil(Node root)
    {
        // Base case : single
        // node satisfies property
        if (root.left == null
            && root.right == null)
            return true;
 
        // node will be in second last level
        if (root.right == null)
        {
            // check heap property at Node
            // No recursive call ,
            // because no need to check last level
            return root.key >= root.left.key;
        }
        else
        {
            // Check heap property at Node and
            // Recursive check heap
            // property at left and
            // right subtree
            if (root.key >= root.left.key
                && root.key >= root.right.key)
                return isHeapUtil(root.left)
                    && isHeapUtil(root.right);
            else
                return false;
        }
    }
 
    // Function to check binary
    // tree is a Heap or Not.
    Boolean isHeap(Node root)
    {
        if (root == null)
            return true;
 
        // These two are used in isCompleteUtil()
        int node_count = countNodes(root);
 
        if (isCompleteUtil(root, 0,
                           node_count) == true
            && isHeapUtil(root) == true)
            return true;
        return false;
    }
 
    // Driver code
    public static void Main(String[] args)
    {
        Is_BinaryTree_MaxHeap bt
            = new Is_BinaryTree_MaxHeap();
 
        Node root = new Node(10);
        root.left = new Node(9);
        root.right = new Node(8);
        root.left.left = new Node(7);
        root.left.right = new Node(6);
        root.right.left = new Node(5);
        root.right.right = new Node(4);
        root.left.left.left = new Node(3);
        root.left.left.right = new Node(2);
        root.left.right.left = new Node(1);
 
        if (bt.isHeap(root) == true)
            Console.WriteLine(
                "Given binary tree is a Heap");
        else
            Console.WriteLine(
                "Given binary tree is not a Heap");
    }
}
 
// This code has been contributed by Arnab Kundu

Javascript




<script>
/* Javascript program to checks if a
binary tree is max heap or not
 */
 
// A Binary Tree node
class Node {
  constructor(k)
  {
    this.key = k;
    this.left = null;
    this.right = null;
  }
}
 
/* This function counts the number
of nodes in a binary tree */
function countNodes(root)
{
    if (root == null)
        return 0;
    return (1 + countNodes(root.left)
            + countNodes(root.right));
}
 
/* This function checks if the
binary tree is complete or not */
function isCompleteUtil(root, index, number_nodes)
{
 
    // An empty tree is complete
    if (root == null)
        return true;
         
    // If index assigned to
    // current node is more than
    // number of nodes in tree, then
    // tree is notcomplete
    if (index >= number_nodes)
        return false;
         
    // Recur for left and right subtrees
    return isCompleteUtil(root.left,
                          2 * index + 1,
                          number_nodes)
        && isCompleteUtil(root.right,
                          2 * index + 2,
                          number_nodes);
}
 
// This Function checks the
// heap property in the tree.
function isHeapUtil(root)
{
    // Base case : single
    // node satisfies property
    if (root.left == null
        && root.right == null)
        return true;
         
    // node will be in second last level
    if (root.right == null)
    {
     
        // check heap property at Node
        // No recursive call ,
        // because no need to check last level
        return root.key >= root.left.key;
    }
    else
    {
     
        // Check heap property at Node and
        // Recursive check heap
        // property at left and
        // right subtree
        if (root.key >= root.left.key
            && root.key >= root.right.key)
            return isHeapUtil(root.left)
                && isHeapUtil(root.right);
        else
            return false;
    }
}
 
// Function to check binary
// tree is a Heap or Not.
function isHeap(root)
{
    if (root == null)
        return true;
         
    // These two are used in isCompleteUtil()
    var node_count = countNodes(root);
    if (isCompleteUtil(root, 0,
                       node_count) == true
        && isHeapUtil(root) == true)
        return true;
    return false;
}
 
// Driver code
var root = new Node(10);
root.left = new Node(9);
root.right = new Node(8);
root.left.left = new Node(7);
root.left.right = new Node(6);
root.right.left = new Node(5);
root.right.right = new Node(4);
root.left.left.left = new Node(3);
root.left.left.right = new Node(2);
root.left.right.left = new Node(1);
if (isHeap(root) == true)
    document.write(
        "Given binary tree is a Heap");
else
    document.write(
        "Given binary tree is not a Heap");
 
// This code is contributed by rrrtnx.
</script>
Output
Given binary tree is a Heap

Method 2: (Iterative approach using level order traversal)

C++




// C++ program to checks if a
// binary tree is max heap or not
#include <bits/stdc++.h>
 
using namespace std;
 
// Tree node structure
struct Node {
    int data;
    struct Node* left;
    struct Node* right;
};
 
// To add a new node
struct Node* newNode(int k)
{
    struct Node* node = new Node;
    node->data = k;
    node->right = node->left = NULL;
    return node;
}
 
bool isHeap(Node* root)
{
    // Your code here
    queue<Node*> q;
    q.push(root);
    bool nullish = false;
    while (!q.empty())
    {
        Node* temp = q.front();
        q.pop();
        if (temp->left)
        {
            if (nullish
                || temp->left->data
                >= temp->data)
            {
                return false;
            }
            q.push(temp->left);
        }
        else {
            nullish = true;
        }
        if (temp->right)
        {
            if (nullish
                || temp->right->data
                >= temp->data)
            {
                return false;
            }
            q.push(temp->right);
        }
        else
        {
            nullish = true;
        }
    }
    return true;
}
 
// Driver code
int main()
{
    struct Node* root = NULL;
    root = newNode(10);
    root->left = newNode(9);
    root->right = newNode(8);
    root->left->left = newNode(7);
    root->left->right = newNode(6);
    root->right->left = newNode(5);
    root->right->right = newNode(4);
    root->left->left->left = newNode(3);
    root->left->left->right = newNode(2);
    root->left->right->left = newNode(1);
 
    // Function call
    if (isHeap(root))
        cout << "Given binary tree is a Heap\n";
    else
        cout << "Given binary tree is not a Heap\n";
 
    return 0;
}

Java




// Java program to checks if a
// binary tree is max heap or not
import java.util.*;
class GFG
{
 
  // Tree node structure
  static class Node
  {
    int data;
    Node left;
    Node right;
  };
 
  // To add a new node
  static Node newNode(int k)
  {
    Node node = new Node();
    node.data = k;
    node.right = node.left = null;
    return node;
  }
 
  static boolean isHeap(Node root)
  {
    Queue<Node> q = new LinkedList<>();
    q.add(root);
    boolean nullish = false;
    while (!q.isEmpty())
    {
      Node temp = q.peek();
      q.remove();
      if (temp.left != null)
      {
        if (nullish
            || temp.left.data
            >= temp.data)
        {
          return false;
        }
        q.add(temp.left);
      }
      else {
        nullish = true;
      }
      if (temp.right != null)
      {
        if (nullish
            || temp.right.data
            >= temp.data)
        {
          return false;
        }
        q.add(temp.right);
      }
      else
      {
        nullish = true;
      }
    }
    return true;
  }
 
  // Driver code
  public static void main(String[] args)
  {
    Node root = null;
    root = newNode(10);
    root.left = newNode(9);
    root.right = newNode(8);
    root.left.left = newNode(7);
    root.left.right = newNode(6);
    root.right.left = newNode(5);
    root.right.right = newNode(4);
    root.left.left.left = newNode(3);
    root.left.left.right = newNode(2);
    root.left.right.left = newNode(1);
 
    // Function call
    if (isHeap(root))
      System.out.print("Given binary tree is a Heap\n");
    else
      System.out.print("Given binary tree is not a Heap\n");
  }
}
 
// This code is contributed by Rajput-Ji

C#




// C# program to checks if a
// binary tree is max heap or not
using System;
using System.Collections.Generic;
public class GFG
{
 
  // Tree node structure
 public
   class Node
  {
    public
 int data;
   public
  Node left;
   public
  Node right;
  };
 
  // To add a new node
  static Node newNode(int k)
  {
    Node node = new Node();
    node.data = k;
    node.right = node.left = null;
    return node;
  }
 
  static bool isHeap(Node root)
  {
    Queue<Node> q = new Queue<Node>();
    q.Enqueue(root);
    bool nullish = false;
    while (q.Count!=0)
    {
      Node temp = q.Peek();
      q.Dequeue();
      if (temp.left != null)
      {
        if (nullish || temp.left.data
            >= temp.data)
        {
          return false;
        }
        q.Enqueue(temp.left);
      }
      else {
        nullish = true;
      }
      if (temp.right != null)
      {
        if (nullish || temp.right.data
            >= temp.data)
        {
          return false;
        }
        q.Enqueue(temp.right);
      }
      else
      {
        nullish = true;
      }
    }
    return true;
  }
 
  // Driver code
  public static void Main(String[] args)
  {
    Node root = null;
    root = newNode(10);
    root.left = newNode(9);
    root.right = newNode(8);
    root.left.left = newNode(7);
    root.left.right = newNode(6);
    root.right.left = newNode(5);
    root.right.right = newNode(4);
    root.left.left.left = newNode(3);
    root.left.left.right = newNode(2);
    root.left.right.left = newNode(1);
 
    // Function call
    if (isHeap(root))
      Console.Write("Given binary tree is a Heap\n");
    else
      Console.Write("Given binary tree is not a Heap\n");
  }
}
 
// This code is contributed by aashish1995

Javascript




<script>
 
    // JavaScript program to checks if a
    // binary tree is max heap or not
     
    class Node
    {
        constructor(data) {
           this.left = null;
           this.right = null;
           this.data = data;
        }
    }
     
    // To add a new node
    function newNode(k)
    {
      let node = new Node(k);
      return node;
    }
 
    function isHeap(root)
    {
      let q = [];
      q.push(root);
      let nullish = false;
      while (q.length > 0)
      {
        let temp = q[0];
        q.shift();
        if (temp.left != null)
        {
          if (nullish
              || temp.left.data
              >= temp.data)
          {
            return false;
          }
          q.push(temp.left);
        }
        else {
          nullish = true;
        }
        if (temp.right != null)
        {
          if (nullish
              || temp.right.data
              >= temp.data)
          {
            return false;
          }
          q.push(temp.right);
        }
        else
        {
          nullish = true;
        }
      }
      return true;
    }
     
    let root = null;
    root = newNode(10);
    root.left = newNode(9);
    root.right = newNode(8);
    root.left.left = newNode(7);
    root.left.right = newNode(6);
    root.right.left = newNode(5);
    root.right.right = newNode(4);
    root.left.left.left = newNode(3);
    root.left.left.right = newNode(2);
    root.left.right.left = newNode(1);
  
    // Function call
    if (isHeap(root))
      document.write("Given binary tree is a Heap" + "</br>");
    else
      document.write("Given binary tree is not a Heap" + "</br>");
     
</script>
Output
Given binary tree is a Heap

This article is contributed by Utkarsh Trivedi. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above




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