# Recursiverly check if a binary tree is height-balanced

In a height-balanced binary tree, the absolute difference of the height of the left sub-tree and the height of the right sub-tree is less than or equal to 1 at every node.

Steps to check if a binary tree is height-balanced are quite straight-forward.

1. Find the height of the left sub-tree and right sub-tree. As we need to find the height of the left and right sub-tree at every node, we use recursion.
2. If the absolute difference between the height of the left sub-tree and the right sub-tree is greater than 1, the tree is not height-balanced.

Note : This is not an efficient method to check if a binary tree is height-balanced as we recursively find the height of the left and the right sub-tree of every node.
Thus every node is visited twice; once while finding if its left and right sub-trees are height-balanced and once when it is a child node whose height needs to be found out.

An efficient approach is to use Tree Traversal which has a time complexity of O(N).

Examples of height balanced and height unbalanced trees

Time complexity : O ( N 2 ), where N is the number of nodes in the tree. We find the height of left and right sub-trees of every node in O(N) time as it is visited. Since there are ‘N’ nodes the time taken is O ( N * N ).

Program to check if a binary tree is height-balanced.

``````class Node:

def __init__(self, data, left = None, right = None):
self.left = left
self.right = right
self.data = data

class Tree:

def FindHeight (self, node):
if ( node == None ):
return 0

return (1 + max(self.FindHeight (node.left), self.FindHeight (node.right)))

def CheckIfHeightBalanced (self, root):

if (root == None):
return True

height_left_subtree = self.FindHeight (root.left)
height_right_subtree = self.FindHeight (root.right)

if (abs(height_left_subtree - height_right_subtree) > 1):
return False

return (self.CheckIfHeightBalanced (root.left) and self.CheckIfHeightBalanced (root.right))

def main():

""" Tree A is height-balanced.
11
/ \ ----- height 0
height 1--- 22

"""
node22 = Node(22)
root_node11 = Node(11, node22, None)

""" Tree B is height-balanced.
1
/ \
height 1----2   3
/ \
4   5 ----- height 2
"""
node2 = Node(2)
node4 = Node(4)
node5 = Node(5)
node3 = Node(3, node4, node5)
root_node1  = Node(1, node2, node3)

""" Tree C is not height-balanced as height difference is abs(1-3) = 2.
10
/ \
height 1--- 20  30
/ \
70  40
/ \
50  60 ------- height 3
"""

node20 = Node(20)
node70 = Node(70)
node50 = Node(50)
node60 = Node(60)
node40 = Node(40, node50, node60)
node30 = Node(30, node70, node40)
root_node10 = Node(10, node20, node30)

roots = [root_node11, root_node1, root_node10]

t = Tree ()
for root in roots :
if (t.CheckIfHeightBalanced(root)) :
print("Tree with root (" + str(root.data) + ") is height balanced.")
else :
print("Tree with root (" + str(root.data) + ") is not height balanced.")

if __name__ == "__main__":
main()
``````

Output

``````Tree with root (11) is height balanced.
Tree with root (1) is height balanced.
Tree with root (10) is not height balanced.
``````
``````#include<iostream>
#include<vector>

using namespace std;

class Node {

public:
int data;
Node * left;
Node * right;
Node(int x) : data(x), left(nullptr), right(nullptr)
{}
Node(int x, Node* left_node, Node* right_node) : data(x), left(left_node), right(right_node)
{}
};

int FindHeight (Node * node) {

if (node == NULL)
return 0;

return (1 + max (FindHeight(node->left), FindHeight(node->right)));
}

bool CheckIfHeightBalanced (Node * root) {

if (root == NULL)
return true;

int height_left_subtree  = FindHeight (root->left);
int height_right_subtree = FindHeight (root->right);

if ( abs(height_left_subtree - height_right_subtree) > 1 ) {
return false;
}

return (CheckIfHeightBalanced(root->left) && CheckIfHeightBalanced(root->right));
}

int main() {

/* Tree A is height-balanced.

11
/ \ ----- height 0
height 1--- 22
*/

Node node22(22);
Node root_node11(11, &node22, nullptr);

/* Tree B is height-balanced.
1
/ \
height 1---- 2   3
/ \
4   5 ----- height 2
*/

Node node4(4), node5(5), node2(2);
Node node3(3, &node4, &node5);
Node root_node1(1, &node2, &node3);

/* Tree C is not height-balanced as height difference is abs(1-3) = 2.
10
/ \
height 1--- 20  30
\
40
/ \
50  60 ------- height 3
*/
Node node50(50), node60(60), node20(20);
Node node40(40, &node50, &node60);
Node node30(30, nullptr, &node40);
Node root_node10(10, &node20, &node30);

vector<Node> roots = { root_node11, root_node1, root_node10 };

for (auto& root : roots) {

if ( CheckIfHeightBalanced (&root) ) {
cout << "Tree with root node (" << root.data << ") is height balanced." << endl;
} else {
cout << "Tree with root node (" << root.data << ") is not height balanced." << endl;
}
}
return 0;
}
``````

Output

``````Tree with root node (11) is height balanced.
Tree with root node (1) is height balanced.
Tree with root node (10) is not height balanced.
``````
``````class Node {

int data;
Node left, right;

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

Node (int n, Node left_child, Node right_child) {
data = n;
left = left_child;
right = right_child;
}

public int FindHeight (Node node) {

if (node == null)
return 0;

return (1 + Math.max(FindHeight(node.left), FindHeight(node.right)));
}

public boolean CheckIfHeightBalanced (Node node) {

if (node == null) {
return Boolean.TRUE;
}

int height_left_subtree = FindHeight(node.left);
int height_right_subtree = FindHeight(node.right);

if (Math.abs(height_left_subtree - height_right_subtree) > 1) {
return Boolean.FALSE;
}

return (CheckIfHeightBalanced(node.left) && CheckIfHeightBalanced(node.right));
}

public static void main (String args[]) {

/* Tree A is height-balanced.
11
/ \ ----- height 0
height 1--- 22
*/

Node root_node11 = new Node(11);
Node node22 = new Node(22);

/* Tree B is height-balanced.
1
/ \
height 1---- 2   3
/ \
4   5 ----- height 2
*/

Node node3 = new Node(3);
Node node4 = new Node(4);
Node node5 = new Node(5);
Node node2 = new Node(2, node4, node5);
Node root_node1 = new Node(1, node2, node3);

/* Tree C is not height-balanced as height difference is abs(1-3) = 2.
10
/ \
height 1 --- 20  30
/ \
70  40
/ \
50  60 ------- height 3
*/

Node node50 = new Node(50);
Node node60 = new Node(60);
Node node70 = new Node(70);
Node node20 = new Node(20);
Node node40 = new Node(40, node50, node60);
Node node30 = new Node(30, node70, node40);
Node root_node10 = new Node(10, node20, node30);

Node [] roots = {root_node11, root_node1, root_node10};

for (Node n : roots) {
if (n.CheckIfHeightBalanced (n)) {
System.out.println("Tree with node (" + n.data + ") is height balanced.");
} else {
System.out.println("Tree with node (" + n.data + ") is not height balanced.");
}
}
}
}
``````

Output

``````Tree with node (11) is height balanced.
Tree with node (1) is height balanced.
Tree with node (10) is not height balanced.
``````