Maximum width of a Binary Tree

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Problem statement

You are given an arbitrary binary tree consisting of N nodes, where each node is associated with a certain value, your task is to find the maximum width of the given tree.

Width of a binary tree is defined as the maximum width of all levels.

For example, consider the following binary tree: example

For the above tree, width of level 1 is 1, width of level 2 is 2, width of level 3 is 3 and width of level 4 is 1. So the maximum width of the tree is 3.

Detailed explanation ( Input/output format, Notes, Images )
Input Format: 
The first line of input contains an integer ‘T’ representing the number of test cases. Then the test cases follow.

The only line of each test case contains values of the nodes in the level order form. The line consists of values of nodes separated by a single space. In case a node is null, we take -1 on 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
Output Format: 
For each test case, a single integer denoting the maximum width of the given binary tree.

The output for each test case is in a separate line.
Note: 
You do not need to print anything; it has already been taken care of.
Constraints: 
1 <= T <= 100
0 <= N <= 3000
-10^6 <= data <= 10^6 and data != -1

Where ‘T’ is the number of test cases, ‘N’ is the total number of nodes in the binary tree, and “data” is the value of the binary tree node.

Time Limit: 1sec
Sample Input 1:
2
5 6 2 4 3 -1 -1 9 7 -1 -1 -1 -1 -1 -1
1 -1 -1
Sample Output 1:
2
1
Explanation of Sample Input 1:

Sample Input 1

For the first test case, the width of 1st level is 1 and for the remaining levels it is 2. So, the maximum width of the binary tree is 2.

For the second test case, the binary tree has only one level having 1 element i.e. root. Hence, the maximum width of the binary tree is 1.
Sample Input 2:
2
1 2 3 4 5 6 7 8 9 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1
1 2 3 4 5 -1 -1 -1 -1 -1 -1
Sample Output 2:
4
2
Hint

Can you think about using level order traversal to count the no. of nodes in each level?

Approaches (3)
Naive Approach

This approach mainly involves two functions. One is to count nodes at a given level (getWidth()), and other is to get the maximum width of the tree (getMaxWidth()). getMaxWidth() makes use of getWidth() to get the width of all levels starting from root.

 

getMaxWidth(): This function returns maximum width of a binary tree. Below is the algorithm:

  • Initialise maxWidth variable with 0.
  • Traverse the binary tree level wise, and for each level
    • Get the width of current level using getWidth().
    • Check if current width is greater than maxWidth so far. If yes, replace maxWidth with current width.
  • Return maxWidth.

getWidth(): This function returns the no. of nodes (width) present at a given level. Below is the algorithm:

  • Check for base cases:
    • If the root is empty, return 0.
    • Return 1, if the current level is 1. As level 1 contains only one node i.e. root.
  • Recursively call to the left and right subtree by decrementing the level variable by 1.
Time Complexity

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

 

Since, for each level out of ‘N’ levels (worst case of Skewed trees), we are calculating the nodes using getWidth() function which takes O(N) time. So, the overall time complexity will be O(N ^ 2).

Space Complexity

O(N), Where ‘N’ is the number of nodes in the given binary tree.

 

We are using O(H) extra space for the call stack where ‘H’ is the height of the tree, and height of a skewed binary tree could be ‘N’ in the worst case.

Code Solution
(100% EXP penalty)
Maximum width of a Binary Tree
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