How to determine if a binary tree is height-balanced?

A tree where no leaf is much farther away from the root than any other leaf. Different balancing schemes allow different definitions of “much farther” and different amounts of work to keep them balanced.
Consider a height-balancing scheme where the following conditions should be checked to determine if a binary tree is balanced. 
An empty tree is height-balanced. A non-empty binary tree T is balanced if: 

1. Left subtree of T is balanced 
2. Right subtree of T is balanced
3. The difference between heights of left subtree and the right subtree is not more than 1. 

The above height-balancing scheme is used in AVL trees. The diagram below shows two trees, one of them is height-balanced and other is not. The second tree is not height-balanced because height of left subtree is 2 more than height of right subtree.

To check if a tree is height-balanced, get the height of left and right subtrees. Return true if difference between heights is not more than 1 and left and right subtrees are balanced, otherwise return false. 

C++

/* CPP program to check if
a tree is height-balanced or not */
#include <bits/stdc++.h>
using namespace std;

/* A binary tree node has data,
pointer to left child and
a pointer to right child */
class node {
public:
    int data;
    node* left;
    node* right;
};

/* Returns the height of a binary tree */
int height(node* node);

/* Returns true if binary tree
with root as root is height-balanced */
bool isBalanced(node* root)
{
    int lh; /* for height of left subtree */
    int rh; /* for height of right subtree */

    /* If tree is empty then return true */
    if (root == NULL)
        return 1;

    /* Get the height of left and right sub trees */
    lh = height(root->left);
    rh = height(root->right);

    if (abs(lh - rh) <= 1 && isBalanced(root->left) && isBalanced(root->right))
        return 1;

    /* If we reach here then
    tree is not height-balanced */
    return 0;
}

/* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */

/* returns maximum of two integers */
int max(int a, int b)
{
    return (a >= b) ? a : b;
}

/* The function Compute the "height"
of a tree. Height is the number of
nodes along the longest path from
the root node down to the farthest leaf node.*/
int height(node* node)
{
    /* base case tree is empty */
    if (node == NULL)
        return 0;

    /* If tree is not empty then
    height = 1 + max of left
        height and right heights */
    return 1 + max(height(node->left),
                   height(node->right));
}

/* Helper function that allocates
a new node with the given data
and NULL left and right pointers. */
node* newNode(int data)
{
    node* Node = new node();
    Node->data = data;
    Node->left = NULL;
    Node->right = NULL;

    return (Node);
}

// Driver code
int main()
{
    node* root = newNode(1);
    root->left = newNode(2);
    root->right = newNode(3);
    root->left->left = newNode(4);
    root->left->right = newNode(5);
    root->left->left->left = newNode(8);

    if (isBalanced(root))
        cout << "Tree is balanced";
    else
        cout << "Tree is not balanced";
    return 0;
}

// This code is contributed by rathbhupendra

C

/* C program to check if a tree is height-balanced or not */
#include <stdio.h>
#include <stdlib.h>
#define bool int

/* A binary tree node has data, pointer to left child
   and a pointer to right child */
struct node {
    int data;
    struct node* left;
    struct node* right;
};

/* Returns the height of a binary tree */
int height(struct node* node);

/* Returns true if binary tree with root as root is height-balanced */
bool isBalanced(struct node* root)
{
    int lh; /* for height of left subtree */
    int rh; /* for height of right subtree */

    /* If tree is empty then return true */
    if (root == NULL)
        return 1;

    /* Get the height of left and right sub trees */
    lh = height(root->left);
    rh = height(root->right);

    if (abs(lh - rh) <= 1 && isBalanced(root->left) && isBalanced(root->right))
        return 1;

    /* If we reach here then tree is not height-balanced */
    return 0;
}

/* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */

/* returns maximum of two integers */
int max(int a, int b)
{
    return (a >= b) ? a : b;
}

/*  The function Compute the "height" of a tree. Height is the
    number of nodes along the longest path from the root node
    down to the farthest leaf node.*/
int height(struct node* node)
{
    /* base case tree is empty */
    if (node == NULL)
        return 0;

    /* If tree is not empty then height = 1 + max of left
      height and right heights */
    return 1 + max(height(node->left), height(node->right));
}

/* Helper function that allocates a new node with the
   given data and NULL left and right pointers. */
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);
}

int main()
{
    struct node* root = newNode(1);
    root->left = newNode(2);
    root->right = newNode(3);
    root->left->left = newNode(4);
    root->left->right = newNode(5);
    root->left->left->left = newNode(8);

    if (isBalanced(root))
        printf("Tree is balanced");
    else
        printf("Tree is not balanced");

    getchar();
    return 0;
}

Java

/* Java program to determine if binary tree is
   height balanced or not */

/* A binary tree node has data, pointer to left child,
   and a pointer to right child */
class Node {
    int data;
    Node left, right;
    Node(int d)
    {
        data = d;
        left = right = null;
    }
}

class BinaryTree {
    Node root;

    /* Returns true if binary tree with root as root is height-balanced */
    boolean isBalanced(Node node)
    {
        int lh; /* for height of left subtree */

        int rh; /* for height of right subtree */

        /* If tree is empty then return true */
        if (node == null)
            return true;

        /* Get the height of left and right sub trees */
        lh = height(node.left);
        rh = height(node.right);

        if (Math.abs(lh - rh) <= 1
            && isBalanced(node.left)
            && isBalanced(node.right))
            return true;

        /* If we reach here then tree is not height-balanced */
        return false;
    }

    /* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */
    /*  The function Compute the "height" of a tree. Height is the
        number of nodes along the longest path from the root node
        down to the farthest leaf node.*/
    int height(Node node)
    {
        /* base case tree is empty */
        if (node == null)
            return 0;

        /* If tree is not empty then height = 1 + max of left
         height and right heights */
        return 1 + Math.max(height(node.left), height(node.right));
    }

    public static void main(String args[])
    {
        BinaryTree tree = new BinaryTree();
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
        tree.root.left.left.left = new Node(8);

        if (tree.isBalanced(tree.root))
            System.out.println("Tree is balanced");
        else
            System.out.println("Tree is not balanced");
    }
}

// This code has been contributed by Mayank Jaiswal(mayank_24)

Python3

"""
Python3 program to check if a tree is height-balanced
"""
# A binary tree Node



class Node:
    # Constructor to create a new Node
    def __init__(self, data):
        self.data = data
        self.left = None
        self.right = None

# function to find height of binary tree
def height(root):

    # base condition when binary tree is empty
    if root is None:
        return 0
    return max(height(root.left), height(root.right)) + 1

# function to check if tree is height-balanced or not
def isBalanced(root):

    # Base condition
    if root is None:
        return True

    # for left and right subtree height
    lh = height(root.left)
    rh = height(root.right)

    # allowed values for (lh - rh) are 1, -1, 0
    if (abs(lh - rh) <= 1) and isBalanced(
    root.left) is True and isBalanced( root.right) is True:
        return True

    # if we reach here means tree is not
    # height-balanced tree
    return False

# Driver function to test the above function
root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
root.left.left.left = Node(8)
if isBalanced(root):
    print("Tree is balanced")
else:
    print("Tree is not balanced")

# This code is contributed by Shweta Singh

C#

using System;

/* C# program to determine if binary tree is
height balanced or not */

/* A binary tree node has data, pointer to left child,
and a pointer to right child */
public class Node {
    public int data;
    public Node left, right;
    public Node(int d)
    {
        data = d;
        left = right = null;
    }
}

public class BinaryTree {
    public Node root;

    /* Returns true if binary tree with root as
    root is height-balanced */
    public virtual bool isBalanced(Node node)
    {
        int lh; // for height of left subtree

        int rh; // for height of right subtree

        /* If tree is empty then return true */
        if (node == null) {
            return true;
        }

        /* Get the height of left and right sub trees */
        lh = height(node.left);
        rh = height(node.right);

        if (Math.Abs(lh - rh) <= 1 && isBalanced(node.left)
            && isBalanced(node.right)) {
            return true;
        }

        /* If we reach here then tree is not height-balanced */
        return false;
    }

    /* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */
    /* The function Compute the "height" of a tree. Height is the
        number of nodes along the longest path from the root node
        down to the farthest leaf node.*/
    public virtual int height(Node node)
    {
        /* base case tree is empty */
        if (node == null) {
            return 0;
        }

        /* If tree is not empty then height = 1 + max of left
        height and right heights */
        return 1 + Math.Max(height(node.left), height(node.right));
    }

    public static void Main(string[] args)
    {
        BinaryTree tree = new BinaryTree();
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
        tree.root.left.left.left = new Node(8);

        if (tree.isBalanced(tree.root)) {
            Console.WriteLine("Tree is balanced");
        }
        else {
            Console.WriteLine("Tree is not balanced");
        }
    }
}

// This code is contributed by Shrikant13

Javascript

<script>

// JavaScript program to check if a tree is height-balanced
//  A binary tree Node

class Node{
    // Constructor to create a new Node
    constructor(data){
        this.data = data
        this.left = null
        this.right = null
    }
}

// function to find height of binary tree
function height(root){

    // base condition when binary tree is empty
    if(root == null)
        return 0
    return Math.max(height(root.left), height(root.right)) + 1
}

// function to check if tree is height-balanced or not
function isBalanced(root){

    // Base condition
    if(root == null)
        return true

    // for left and right subtree height
    let lh = height(root.left)
    let rh = height(root.right)

    // allowed values for (lh - rh) are 1, -1, 0
    if (Math.abs(lh - rh) <= 1 && isBalanced(
    root.left)== true && isBalanced( root.right) == true)
        return true

    // if we reach here means tree is not
    // height-balanced tree
    return false
}

// Driver function to test the above function
let root = new Node(1)
root.left = new Node(2)
root.right = new Node(3)
root.left.left = new Node(4)
root.left.right = new Node(5)
root.left.left.left = new Node(8)
if(isBalanced(root))
    document.write("Tree is balanced","</br>")
else
    document.write("Tree is not balanced","</br>")

// This code is contributed by ShinjanPatra

</script>

Output:

Tree is not balanced

Time Complexity: O(n^2) in case of full binary tree.

Auxiliary Space: O(n) space for call stack since using recursion

Optimized implementation: Above implementation can be optimized by calculating the height in the same recursion rather than calling a height() function separately. Thanks to Amar for suggesting this optimized version. This optimization reduces time complexity to O(n).

C++

/* C++ program to check if a tree
is height-balanced or not */
#include <bits/stdc++.h>
using namespace std;
#define bool int

/* A binary tree node has data,
pointer to left child and
a pointer to right child */
class node {
public:
    int data;
    node* left;
    node* right;
};

/* The function returns true if root is
balanced else false The second parameter
is to store the height of tree. Initially,
we need to pass a pointer to a location with
value as 0. We can also write a wrapper
over this function */
bool isBalanced(node* root, int* height)
{

    /* lh --> Height of left subtree
    rh --> Height of right subtree */
    int lh = 0, rh = 0;

    /* l will be true if left subtree is balanced
    and r will be true if right subtree is balanced */
    int l = 0, r = 0;

    if (root == NULL) {
        *height = 0;
        return 1;
    }

    /* Get the heights of left and right subtrees in lh and rh
    And store the returned values in l and r */
    l = isBalanced(root->left, &lh);
    r = isBalanced(root->right, &rh);

    /* Height of current node is max of heights of left and
    right subtrees plus 1*/
    *height = (lh > rh ? lh : rh) + 1;

    /* If difference between heights of left and right
    subtrees is more than 2 then this node is not balanced
    so return 0 */
    if (abs(lh - rh) >= 2)
        return 0;

    /* If this node is balanced and left and right subtrees
    are balanced then return true */
    else
        return l && r;
}

/* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */

/* Helper function that allocates a new node with the
given data and NULL left and right pointers. */
node* newNode(int data)
{
    node* Node = new node();
    Node->data = data;
    Node->left = NULL;
    Node->right = NULL;

    return (Node);
}

// Driver code
int main()
{
    int height = 0;

    /* Constructed binary tree is
            1
            / \
            2 3
            / \ /
            4 5 6
            /
            7
    */
    node* 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);
    root->left->left->left = newNode(7);

    if (isBalanced(root, &height))
        cout << "Tree is balanced";
    else
        cout << "Tree is not balanced";

    return 0;
}

// This is code is contributed by rathbhupendra

C

/* C program to check if a tree is height-balanced or not */
#include <stdio.h>
#include <stdlib.h>
#define bool int

/* A binary tree node has data, pointer to left child
   and a pointer to right child */
struct node {
    int data;
    struct node* left;
    struct node* right;
};

/* The function returns true if root is balanced else false
   The second parameter is to store the height of tree.
   Initially, we need to pass a pointer to a location with value
   as 0. We can also write a wrapper over this function */
bool isBalanced(struct node* root, int* height)
{
    /* lh --> Height of left subtree
     rh --> Height of right subtree */
    int lh = 0, rh = 0;

    /* l will be true if left subtree is balanced
    and r will be true if right subtree is balanced */
    int l = 0, r = 0;

    if (root == NULL) {
        *height = 0;
        return 1;
    }

    /* Get the heights of left and right subtrees in lh and rh
    And store the returned values in l and r */
    l = isBalanced(root->left, &lh);
    r = isBalanced(root->right, &rh);

    /* Height of current node is max of heights of left and
     right subtrees plus 1*/
    *height = (lh > rh ? lh : rh) + 1;

    /* If difference between heights of left and right
     subtrees is more than 2 then this node is not balanced
     so return 0 */
    if (abs(lh - rh) >= 2)
        return 0;

    /* If this node is balanced and left and right subtrees
    are balanced then return true */
    else
        return l && r;
}

/* UTILITY FUNCTIONS TO TEST isBalanced() FUNCTION */

/* Helper function that allocates a new node with the
   given data and NULL left and right pointers. */
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);
}

// Driver code
int main()
{
    int height = 0;

    /* Constructed binary tree is
             1
           /   \
         2      3
       /  \    /
     4     5  6
    /
   7
  */
    struct node* 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);
    root->left->left->left = newNode(7);

    if (isBalanced(root, &height))
        printf("Tree is balanced");
    else
        printf("Tree is not balanced");

    getchar();
    return 0;
}

Java

/* Java program to determine if binary tree is
   height balanced or not */

/* A binary tree node has data, pointer to left child,
   and a pointer to right child */
class Node {

    int data;
    Node left, right;

    Node(int d)
    {
        data = d;
        left = right = null;
    }
}

// A wrapper class used to modify height across
// recursive calls.
class Height {
    int height = 0;
}

class BinaryTree {

    Node root;

    /* Returns true if binary tree with root as root is height-balanced */
    boolean isBalanced(Node root, Height height)
    {
        /* If tree is empty then return true */
        if (root == null) {
            height.height = 0;
            return true;
        }

        /* Get heights of left and right sub trees */
        Height lheight = new Height(), rheight = new Height();
        boolean l = isBalanced(root.left, lheight);
        boolean r = isBalanced(root.right, rheight);
        int lh = lheight.height, rh = rheight.height;

        /* Height of current node is max of heights of
           left and right subtrees plus 1*/
        height.height = (lh > rh ? lh : rh) + 1;

        /* If difference between heights of left and right
           subtrees is more than 2 then this node is not balanced
           so return 0 */
        if (Math.abs(lh - rh) >= 2)
            return false;

        /* If this node is balanced and left and right subtrees
           are balanced then return true */
        else
            return l && r;
    }

    public static void main(String args[])
    {
        Height height = new Height();

        /* Constructed binary tree is
                   1
                 /   \
                2      3
              /  \    /
            4     5  6
            /
           7         */
        BinaryTree tree = new BinaryTree();
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
        tree.root.right.right = new Node(6);
        tree.root.left.left.left = new Node(7);

        if (tree.isBalanced(tree.root, height))
            System.out.println("Tree is balanced");
        else
            System.out.println("Tree is not balanced");
    }
}

// This code has been contributed by Mayank Jaiswal(mayank_24)

Python3

"""
Python3 program to check if Binary tree is
height-balanced
"""

# A binary tree node
class Node:

    # constructor to create node of
    # binary tree
    def __init__(self, data):
        self.data = data
        self.left = self.right = None

# utility class to pass height object
class Height:
    def __init__(self):
        self.height = 0

# function to find height of binary tree
def height(root):

    # base condition when binary tree is empty
    if root is None:
        return 0
    return max(height(root.left), height(root.right)) + 1

# helper function to check if binary
# tree is height balanced
def isBalanced(root):
    # Base condition when tree is
    # empty return true
    if root is None:
        return True

    # lh and rh to store height of
    # left and right subtree
    lh = Height()
    rh = Height()

    # Calculating height of left and right tree
    lh.height = height(root.left)
    rh.height = height(root.right)


    # l and r are used to check if left
    # and right subtree are balanced
    l = isBalanced(root.left)
    r = isBalanced(root.right)

    # height of tree is maximum of
    # left subtree height and
    # right subtree height plus 1

    if abs(lh.height - rh.height) <= 1:
        return l and r

    # if we reach here then the tree
    # is not balanced
    return False

# Driver function to test the above function
"""
Constructed binary tree is
            1
        / \
        2     3
    / \ /
    4 5 6 / 7
"""
# to store the height of tree during traversal

root = Node(1)
root.left = Node(2)
root.right = Node(3)
root.left.left = Node(4)
root.left.right = Node(5)
root.right.left = Node(6)
root.left.left.left = Node(7)

if isBalanced(root):
    print('Tree is balanced')
else:
    print('Tree is not balanced')

# This code is contributed by Shubhank Gupta

C#

using System;

/* C# program to determine if binary tree is
   height balanced or not */

/* A binary tree node has data, pointer to left child,
   and a pointer to right child */
public class Node {

    public int data;
    public Node left, right;

    public Node(int d)
    {
        data = d;
        left = right = null;
    }
}

// A wrapper class used to modify height across
// recursive calls.
public class Height {
    public int height = 0;
}

public class BinaryTree {

    public Node root;

    /* Returns true if binary tree with root as root is height-balanced */
    public virtual bool isBalanced(Node root, Height height)
    {
        /* If tree is empty then return true */
        if (root == null) {
            height.height = 0;
            return true;
        }

        /* Get heights of left and right sub trees */
        Height lheight = new Height(), rheight = new Height();
        bool l = isBalanced(root.left, lheight);
        bool r = isBalanced(root.right, rheight);
        int lh = lheight.height, rh = rheight.height;

        /* Height of current node is max of heights of
           left and right subtrees plus 1*/
        height.height = (lh > rh ? lh : rh) + 1;

        /* If difference between heights of left and right
           subtrees is more than 2 then this node is not balanced
           so return 0 */
        if (Math.Abs(lh - rh) >= 2) {
            return false;
        }

        /* If this node is balanced and left and right subtrees
           are balanced then return true */
        else {
            return l && r;
        }
    }

    /*  The function Compute the "height" of a tree. Height is the
        number of nodes along the longest path from the root node
        down to the farthest leaf node.*/
    public virtual int height(Node node)
    {
        /* base case tree is empty */
        if (node == null) {
            return 0;
        }

        /* If tree is not empty then height = 1 + max of left
         height and right heights */
        return 1 + Math.Max(height(node.left), height(node.right));
    }

    public static void Main(string[] args)
    {
        Height height = new Height();

        /* Constructed binary tree is
                   1
                 /   \
                2      3
              /  \    /
            4     5  6
            /
           7         */
        BinaryTree tree = new BinaryTree();
        tree.root = new Node(1);
        tree.root.left = new Node(2);
        tree.root.right = new Node(3);
        tree.root.left.left = new Node(4);
        tree.root.left.right = new Node(5);
        tree.root.right.right = new Node(6);
        tree.root.left.left.left = new Node(7);

        if (tree.isBalanced(tree.root, height)) {
            Console.WriteLine("Tree is balanced");
        }
        else {
            Console.WriteLine("Tree is not balanced");
        }
    }
}

// This code is contributed by Shrikant13

Javascript

<script>

// JavaScript program to check if Binary tree is height-balanced

// A binary tree node
class Node{

    // constructor to create node of
    // binary tree
    constructor(data){
        this.data = data
        this.left = this.right = null
    }
}

// utility class to pass height object
class Height
{
    constructor()
    {
        this.height = 0
    }
}

// function to find height of binary tree
function height(root){

    // base condition when binary tree is empty
    if(root == null)
        return 0
    return Math.max(height(root.left), height(root.right)) + 1

}

// helper function to check if binary
// tree is height balanced
function isBalanced(root)
{

    // Base condition when tree is
    // empty return true
    if(root == null)
        return true

    // lh and rh to store height of
    // left and right subtree
    let lh = new Height()
    let rh = new Height()

    // Calculating height of left and right tree
    lh.height = height(root.left)
    rh.height = height(root.right)


    // l and r are used to check if left
    // and right subtree are balanced
    let l = isBalanced(root.left)
    let r = isBalanced(root.right)

    // height of tree is maximum of
    // left subtree height and
    // right subtree height plus 1

    if(Math.abs(lh.height - rh.height) <= 1)
        return l && r

    // if we reach here then the tree
    // is not balanced
    return false

}

// Driver function to test the above function

// Constructed binary tree is
//             1
//         / \
//         2     3
//     / \ /
//     4 5 6 / 7
//
// to store the height of tree during traversal

let root = new Node(1)
root.left = new Node(2)
root.right = new Node(3)
root.left.left = new Node(4)
root.left.right = new Node(5)
root.right.left = new Node(6)
root.left.left.left = new Node(7)

if(isBalanced(root))
    document.write('Tree is balanced',"</br>")
else
    document.write('Tree is not balanced',"</br>")

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

Output:

Tree is balanced

Time Complexity: O(n)
Auxiliary Space: O(n)

https://www.youtube.com/watch?v=qIdVVH1tKM4

Asked in: Amazon, Belzabar, Goldman Sachs, InMobi, Intel, Microsoft, Paytm, Synopsys, Walmart, Zillious

Please write comments if you find any of the above codes/algorithms incorrect, or find other ways to solve the same problem.