Check if a given Binary Tree is a Heap

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

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

Examples:

Input:

Output: Given binary tree is a heap 

Input: 

Output: Given binary tree is not a heap

Check if a given Binary Tree is Heap using Complete Binary Tree

Follow the given steps to solve the problem:

• Check each of the above conditions separately, for checking completeness isComplete and for checking heap isHeapUtil functions are written. 
• First, check if the given binary tree is complete or not.
• Then to check if the binary tree is a heap or not, check the following points:
  • Every Node has 2 children, 0 children (last level nodes), or 1 child (there can be at most one such node).
  • If Node has No children then it’s a leaf node and returns true (Base case)
  • If Node has one child (it must be the left child because it is a complete tree) then compare this node with its single child only.
  • If the Node has both children then check heap property at this Node and recur for both subtrees. 

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's 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's 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))
        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

Python3

# Python3 code 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_property_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_property_util(root.left) and
                        self.heap_property_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_property_util(self)):
            return True
        else:
            return False


# Driver's Code
if __name__ == '__main__':
    root = GFG(5)
    root.left = GFG(2)
    root.right = GFG(3)
    root.left.left = GFG(1)

    # Function call
    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's 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);

        // Function call
        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

/* 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's 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);

// Function call
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.

Output

Given binary tree is a Heap

Time Complexity: O(N), where N is the number of nodes
Auxiliary Space: O(logN), for recursive stack space.

Check if a given Binary Tree is Heap using Level Order Traversal:

Level order traversal can be used to check heap properties at each level of the binary tree. Check whether value of each node is greater than the value of its children and keep track of when the last node is encountered and whether it is following the heap properties using a boolean flag 

Follow the given steps to solve the problem:

• declare a queue for level order traversal and a flag variable nullish equal to false
• Start level order traversal 
  • Check for the left child of the node and if either the nullish is true or root’s value is less than its left child node, then return false, else push this node into the queue
  • If the node’s left child is null then set nullish equal to true, which means we have already encountered the last node, as the node with only zero or one children can occur only once in the complete tree
  • Now check the right child of the node and if either the nullish is true or root’s value is less than its right child node, then return false, else push this node into the queue.
  • If the node’s right child is null then set nullish equal to true, which means we have already encountered the last node, as the node with only zero or one children can occur only once in the complete tree
• Return true after checking every node in the 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's 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's 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

Python3

# Python3 program to check if a binary tree is max heap or not.

from collections import deque


class Node:
    def __init__(self, value):
        self.key = value
        self.left = None
        self.right = None


def isHeap(root):
    queue = deque()
    queue.append(root)
    nullish = False
    while len(queue) > 0:
        temp = queue[0]
        queue.popleft()
        if temp.left:
            if nullish or temp.left.key > temp.key:
                return False
            queue.append(temp.left)
        else:
            nullish = True
        if temp.right:
            if nullish or temp.right.key > temp.key:
                return False
            queue.append(temp.right)
        else:
            nullish = True
    return True


# Driver's code
if __name__ == '__main__':
    root = Node(10)
    root.left = Node(9)
    root.right = Node(8)
    root.left.left = Node(7)
    root.left.right = Node(6)
    root.right.left = Node(5)
    root.right.right = Node(4)
    root.left.left.left = Node(3)
    root.left.left.right = Node(2)
    root.left.right.left = Node(1)

    # Function call
    if isHeap(root):
        print("Given binary tree is a Heap")
    else:
        print("Given binary tree is not a Heap")

# This code is contributed by lokeshmvs21.

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's 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

// 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>");

Output

Given binary tree is a Heap

Time Complexity: O(N) where N is the total number of nodes in a given binary tree.
Auxiliary Space: O(N)

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