Maximum Height Difference

Easy
0/40
Average time to solve is 15m
2 upvotes
Asked in companies
Morgan StanleyYatra

Problem statement

You are given the root node of a binary tree consisting of 'N' nodes. Your task is to return the maximum height difference of the tree.

The height of a Binary Tree is defined as the number of nodes present in the longest path from the root node to any leaf node of the tree. The height difference of a node is equal to the absolute difference of height of the left and right subtree.

For example:

Example

For the given tree,
The maximum height difference is 1. The height difference of node 1 is the absolute difference of the height of subtree with root node 2, and the height of subtree with root node 3, which is 1. Hence the answer is 1.
Example: 
Elements are in the level order form. The input consists of values of nodes separated by a single space in a single line. In case a node is null, we take -1 in its place.

For example, the input for the tree depicted in the below image would be :

Example

1
2 3
4 -1 5 6
-1 7 -1 -1 -1 -1
-1 -1

Explanation :
Level 1 :
The root node of the tree is 1

Level 2 :
Left child of 1 = 2
Right child of 1 = 3

Level 3 :
Left child of 2 = 4
Right child of 2 = null (-1)
Left child of 3 = 5
Right child of 3 = 6

Level 4 :
Left child of 4 = null (-1)
Right child of 4 = 7
Left child of 5 = null (-1)
Right child of 5 = null (-1)
Left child of 6 = null (-1)
Right child of 6 = null (-1)

Level 5 :
Left child of 7 = null (-1)
Right child of 7 = null (-1)

The first not-null node (of the previous level) is treated as the parent of the first two nodes of the current level. 

The second not-null node (of the previous level) is treated as the parent node for the next two nodes of the current level and so on.

The input ends when all nodes at the last level are null (-1).
Note : 
The above format was just to provide clarity on how the input is formed for a given tree. 

The sequence will be put together in a single line separated by a single space. Hence, for the above-depicted tree, the input will be given as:

1 2 3 4 -1 5 6 -1 7 -1 -1 -1 -1 -1 -1
Detailed explanation ( Input/output format, Notes, Images )
Input Format: 
The first line of the input contains an integer, 'T,’ denoting the number of test cases.

The first line of each test case contains the elements of the tree in the level order form separated by a single space. If any node does not have a left or right child, take -1 in its place. Refer to the example for further clarification.
Output Format: 
For each test case, print a single integer that denotes the maximum height difference for the given tree.

Print the output of each test case in a separate line.
Note: 
You do not need to print anything. It has already been taken care of. Just implement the given function.
Constraints: 
1 <= T <= 10
1 <= N <= 10^6
1 <= nodeVal <=10^9

Time limit: 1 sec
Sample Input 1:
2
1 2 3 4 5 -1 -1 -1 -1 -1 -1 
1 -1 2 -1 3 -1 4 -1 -1
Sample Output 1:
1
3
Explanation of sample input 1:
For the first test case, 
The maximum height difference is 1. The height difference of node 1 is the absolute difference of the height of subtree with root node 2, and the height of subtree with root node 3, which is 1. Hence the answer is 1.

Example

For the second test case,
The maximum height difference is 3. The height difference of node 1 is the absolute difference of the height of subtree with root node 2, and the height of subtree with an empty node, which is 3. Hence the answer is 3.
Sample Input 2:
2
1 2 3 4 -1 7 8 5 -1 9 -1 -1 -1 6 -1 -1 -1 -1 -1
10 20 11 -1 -1 -1 -1
Sample Output 2:
3
0
Hint

Can we create a function to calculate the height of a node?

Approaches (3)
Brute Force

This approach will create a function height(node) that will calculate the height of the node.

We will travel through all nodes, call the height function, and calculate the height difference between the left and right subtree and update the maximum height difference of the tree. We will traverse all nodes using traverse(node) function.

 

Algorithm:

  • Defining the height(node) function:
    • If the node is the empty node, return 0.
    • Now we will recursively call the height functions for both left and right children. Define two variables leftHeight and rightHeight to store the height of left and right subtrees.
    • Set leftHeight as height(left subtree of the node).
    • Set rightHeight as height(right subtree of the node).
    • curHeight is a variable to store the value of the current node.
    • Set curHeight as the maximum of leftHeight and rightHeight. 
    • We will increment curHeight by 1.
    • Return curHeight.

 

  • We will traverse the tree using a function traverse(node), which returns the maximum height difference in the subtree whose root is node.
  • Defining traverse(node) function:
    • If node is an empty node, return 0.
    • Set lHeight as height(left child of the node).
    • Set rHeight as height(right child of the node).
    • Declare a variable curDiff to store the height difference of the current node.
    • Set curDiff as absolute difference of lHeight and rHeight.
    • Now traverse to the left and right child.
    • Declare maxDiffLeft and maxDiffRight to store the maximum height difference in the left subtree and right subtree.
    • Set maxDiffLeft as traverse(left child of node).
    • Set maxDiffRight as traverse(right child of node).
    • Set maxDiff as the maximum of maxDiffLeft, maxDiffRight, and curDiff.
    • Return maxDiff.

 

  • Store the maximum height difference of the tree in a variable ans.
  • Set ans as traverse(root)
  • Return the variable ans.
Time Complexity

O(N^2), where N is the number of nodes in the tree.

 

In at worst case(For skew trees), the time complexity of height function will be O(N). We have to travel all the N nodes to calculate the height of the root node. We call the height function to find the height of the left and right subtree for each node. So the time complexity will be N*N. Hence, the overall time complexity is O(N^2).

Space Complexity

O(N), where N is the number of nodes in the tree.

 

We use O(N) space complexity for recursive stack during recursion calls of height and traverse function in the worst case. Hence the overall space complexity is O(N).

Code Solution
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Maximum Height Difference
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