How to check if a given array represents a Binary Heap?

Given an array, how to check if the given array represents a Binary Max-Heap.
Examples:

Input:  arr[] = {90, 15, 10, 7, 12, 2} 
Output: True
The given array represents below tree
       90
     /    \
   15      10
  /  \     /
 7    12  2 
The tree follows max-heap property as every
node is greater than all of its descendants.

Input:  arr[] = {9, 15, 10, 7, 12, 11} 
Output: False
The given array represents below tree
       9
     /    \
   15      10
  /  \     /
 7    12  11
The tree doesn't follows max-heap property 9 is 
smaller than 15 and 10, and 10 is smaller than 11. 

A Simple Solution is to first check root if it’s greater than all of its descendants. Then check for children of the root. Time complexity of this solution is O(n²)

An Efficient Solution is to compare root only with its children (not all descendants), if root is greater than its children and the same is true for all nodes, then tree is max-heap (This conclusion is based on transitive property of > operator, i.e., if x > y and y > z, then x > z).
The last internal node is present at index (n-2)/2 assuming that indexing begins with 0.

Below is the implementation of this solution. 

C++

// C program to check whether a given array
// represents a max-heap or not
#include <limits.h>
#include <stdio.h>

// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[], int i, int n)
{
    // If (2 * i) + 1 >= n, then leaf node, so return true
    if (i >= (n - 1) / 2)
        return true;

    // If an internal node and is
    // greater than its children,
    // and same is recursively
    // true for the children
    if (arr[i] >= arr[2 * i + 1] &&
        arr[i] >= arr[2 * i + 2]
        && isHeap(arr, 2 * i + 1, n)
        && isHeap(arr, 2 * i + 2, n))
        return true;

    return false;
}

// Driver program
int main()
{
    int arr[] = { 90, 15, 10, 7, 12, 2, 7, 3 };
    int n = sizeof(arr) / sizeof(int) - 1;

    isHeap(arr, 0, n) ? printf("Yes") : printf("No");

    return 0;
}

Java

// Java program to check whether a given array
// represents a max-heap or not
class GFG
{

    // Returns true if arr[i..n-1]
    // represents a max-heap
    static boolean isHeap(int arr[],
                          int i, int n)
    {
        // If (2 * i) + 1 >= n, then leaf node, so return true
        if (i >= (n - 1) / 2)
        {
            return true;
        }

        // If an internal node and
        // is greater than its
        // children, and same is
        // recursively true for the
        // children
        if (arr[i] >= arr[2 * i + 1]
            && arr[i] >= arr[2 * i + 2]
            && isHeap(arr, 2 * i + 1, n)
            && isHeap(arr, 2 * i + 2, n))
        {
            return true;
        }

        return false;
    }

    // Driver program
    public static void main(String[] args)
    {
        int arr[] = { 90, 15, 10, 7, 12, 2, 7, 3 };
        int n = arr.length - 1;
        if (isHeap(arr, 0, n)) {
            System.out.println("Yes");
        }
        else {
            System.out.println("No");
        }
    }
}

// This code contributed by 29AjayKumar

Python3

# Python3 program to check whether a
# given array represents a max-heap or not

# Returns true if arr[i..n-1]
# represents a max-heap
def isHeap(arr, i, n):

    # If (2 * i) + 1 >= n, then leaf node, so return true
    if i >= int((n - 1) / 2):
        return True

    # If an internal node and is greater
    # than its children, and same is
    # recursively true for the children
    if(arr[i] >= arr[2 * i + 1] and
       arr[i] >= arr[2 * i + 2] and
       isHeap(arr, 2 * i + 1, n) and
       isHeap(arr, 2 * i + 2, n)):
        return True

    return False

# Driver Code
if __name__ == '__main__':
    arr = [90, 15, 10, 7, 12, 2, 7, 3]
    n = len(arr) - 1

    if isHeap(arr, 0, n):
        print("Yes")
    else:
        print("No")

# This code is contributed by PranchalK

C#

// C# program to check whether a given
// array represents a max-heap or not
using System;

class GFG
{

// Returns true if arr[i..n-1] represents a
// max-heap
static bool isHeap(int []arr, int i, int n)
{
    // If (2 * i) + 1 >= n, then leaf node, so return true
    if (i >= (n - 1) / 2)
    {
        return true;
    }

    // If an internal node and is greater
    // than its children, and same is
    // recursively true for the children
    if (arr[i] >= arr[2 * i + 1] &&
        arr[i] >= arr[2 * i + 2] &&
        isHeap(arr, 2 * i + 1, n) &&
        isHeap(arr, 2 * i + 2, n))
    {
        return true;
    }

    return false;
}

// Driver Code
public static void Main(String[] args)
{
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = arr.Length-1;
    if (isHeap(arr, 0, n))
    {
        Console.Write("Yes");
    }

    else
    {
        Console.Write("No");
    }
}
}

PHP

<?php
// PHP program to check whether a given
// array represents a max-heap or not

// Returns true if arr[i..n-1]
// represents a max-heap
function isHeap($arr, $i, $n)
{

// If (2 * i) + 1 >= n, then leaf node, so return true
if ($i >= ($n - 1) / 2)
    return true;

// If an internal node and is greater
// than its children, and same is
// recursively true for the children
if ($arr[$i] >= $arr[2 * $i + 1] &&
    $arr[$i] >= $arr[2 * $i + 2] &&
    isHeap($arr, 2 * $i + 1, $n) &&
    isHeap($arr, 2 * $i + 2, $n))
    return true;

return false;
}

// Driver Code
$arr = array(90, 15, 10, 7, 12, 2, 7, 3);
$n = sizeof($arr);

if(isHeap($arr, 0, $n))
    echo "Yes";
else
    echo "No";

// This code is contributed
// by Akanksha Rai
?>

Javascript

<script>
// Javascript program to check whether a given array
// represents a max-heap or not

// Returns true if arr[i..n-1]
    // represents a max-heap
function isHeap(arr,i,n)
{
    // If (2 * i) + 1 >= n, then leaf node, so return true
        if (i >= (n - 1) / 2)
        {
            return true;
        }

        // If an internal node and
        // is greater than its
        // children, and same is
        // recursively true for the
        // children
        if (arr[i] >= arr[2 * i + 1]
            && arr[i] >= arr[2 * i + 2]
            && isHeap(arr, 2 * i + 1, n)
            && isHeap(arr, 2 * i + 2, n))
        {
            return true;
        }

        return false;
}

// Driver program
let arr=[ 90, 15, 10, 7, 12, 2, 7, 3 ];
let n = arr.length - 1;
if (isHeap(arr, 0, n)) {
    document.write("Yes<br>");
}
else {
    document.write("No<br>");
}


// This code is contributed by rag2127
</script>

Output

Yes

Time complexity: O(n)
Auxiliary Space: O(h), Here h is the height of the given tree and the extra space is used due to the recursion call stack.

An Iterative Solution is to traverse all internal nodes and check id the node is greater than its children or not. 

C++

// C program to check whether a given array
// represents a max-heap or not
#include <stdio.h>
#include <limits.h>

// Returns true if arr[i..n-1] represents a
// max-heap
bool isHeap(int arr[],  int n)
{
    // Start from root and go till the last internal
    // node
    for (int i=0; i<=(n-2)/2; i++)
    {
        // If left child is greater, return false
        if (arr[2*i +1] > arr[i])
                return false;

        // If right child is greater, return false
        if (2*i+2 < n && arr[2*i+2] > arr[i])
                return false;
    }
    return true;
}

// Driver program
int main()
{
    int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = sizeof(arr) / sizeof(int);

    isHeap(arr, n)? printf("Yes"): printf("No");

    return 0;
}

Java

// Java program to check whether a given array
// represents a max-heap or not

class GFG {

// Returns true if arr[i..n-1] represents a
// max-heap
    static boolean isHeap(int arr[], int n) {
        // Start from root and go till the last internal
        // node
        for (int i = 0; i <= (n - 2) / 2; i++) {
            // If left child is greater, return false
            if (arr[2 * i + 1] > arr[i]) {
                return false;
            }

            // If right child is greater, return false
            if (2 * i + 2 < n && arr[2 * i + 2] > arr[i]) {
                return false;
            }
        }
        return true;
    }

// Driver program
    public static void main(String[] args) {
        int arr[] = {90, 15, 10, 7, 12, 2, 7, 3};
        int n = arr.length;
        if (isHeap(arr, n)) {
            System.out.println("Yes");
        } else {
            System.out.println("No");
        }
    }
}
// This code is contributed by 29AjayKumar

Python3

# Python3 program to check whether a
# given array represents a max-heap or not

# Returns true if arr[i..n-1]
# represents a max-heap
def isHeap(arr, n):

    # Start from root and go till
    # the last internal node
    for i in range(int((n - 2) / 2) + 1):

        # If left child is greater,
        # return false
        if arr[2 * i + 1] > arr[i]:
                return False

        # If right child is greater,
        # return false
        if (2 * i + 2 < n and
            arr[2 * i + 2] > arr[i]):
                return False
    return True

# Driver Code
if __name__ == '__main__':
    arr = [90, 15, 10, 7, 12, 2, 7, 3]
    n = len(arr)

    if isHeap(arr, n):
        print("Yes")
    else:
        print("No")

# This code is contributed by PranchalK

C#

// C# program to check whether a given array
// represents a max-heap or not
using System;

class GFG
{

// Returns true if arr[i..n-1]
// represents a max-heap
static bool isHeap(int []arr, int n)
{
    // Start from root and go till
    // the last internal node
    for (int i = 0; i <= (n - 2) / 2; i++)
    {
        // If left child is greater,
        // return false
        if (arr[2 * i + 1] > arr[i])
        {
            return false;
        }

        // If right child is greater,
        // return false
        if (2 * i + 2 < n && arr[2 * i + 2] > arr[i])
        {
            return false;
        }
    }
    return true;
}

// Driver Code
public static void Main()
{
    int []arr = {90, 15, 10, 7, 12, 2, 7, 3};
    int n = arr.Length;
    if (isHeap(arr, n))
    {
        Console.Write("Yes");
    }
    else
    {
        Console.Write("No");
    }
}
}

// This code is contributed
// by 29AjayKumar

PHP

<?php
// PHP program to check whether a
// given array represents a max-heap or not

// Returns true if arr[i..n-1]
// represents a max-heap
function isHeap($arr, $i, $n)
{
    // Start from root and go till
    // the last internal node
    for ($i = 0; $i < (($n - 2) / 2) + 1; $i++)
    {
        // If left child is greater,
        // return false
        if($arr[2 * $i + 1] > $arr[$i])
                return False;

        // If right child is greater,
        // return false
        if (2 * $i + 2 < $n &&
                $arr[2 * $i + 2] > $arr[$i])
                return False;

    return True;
    }
}

// Driver Code
$arr = array(90, 15, 10, 7, 12, 2, 7, 3);
$n = sizeof($arr);

if(isHeap($arr, 0, $n))
    echo "Yes";
else
    echo "No";

// This code is contributed by Princi Singh
?>

Javascript

<script>

// Javascript program to check
// whether a given array
// represents a max-heap or not

// Returns true if arr[i..n-1]
// represents a max-heap
function isHeap( arr, n)
{
    // Start from root and go till
    // the last internal node
    for (let i=0; i<=Math.floor((n-2)/2); i++)
    {
        // If left child is greater,
        // return false
        if (arr[2*i +1] > arr[i])
                return false;

        // If right child is greater,
        // return false
        if (2*i+2 < n && arr[2*i+2] > arr[i])
                return false;
    }
    return true;
}
    // driver code

    let arr = [90, 15, 10, 7, 12, 2, 7, 3];
    let n = arr.length;
    if (isHeap(arr, n)) {
        document.write("Yes");
        }
    else {
        document.write("No");
        }

</script>

Output

Yes

Time complexity: O(n), Where n is the total number of elements in the given array.
Auxiliary Space: O(1), As constant extra space is used.

Thanks to Himanshu for suggesting this solution.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above