Maximum Width of Binary Tree

Maximum Width of Binary Tree

What is this problem about?

The "Maximum Width of Binary Tree" coding problem asks you to find the maximum width among all levels of a given binary tree. The width of a level is defined as the length between the leftmost and rightmost non-null nodes at that level, where the null nodes between them are also counted. For instance, if you imagine the nodes being placed in an array-like structure where the root is at index 0, its children are at 1 and 2, their children at 3, 4, 5, 6, and so on, then the width is rightmost_index - leftmost_index + 1. This problem tests your ability to traverse binary trees efficiently and accurately track node positions across levels.

Why is this asked in interviews?

This Maximum Width of Binary Tree interview question is a popular choice for companies like Apple, Meta, and Google because it effectively assesses a candidate's understanding of tree traversal algorithms, particularly Breadth-First Search (BFS) and Depth-First Search (DFS). It challenges you to assign conceptual positions to nodes, including null nodes, to correctly calculate width. Interviewers want to see how you manage level-by-level processing (BFS) or how you maintain state during a recursive traversal (DFS) to find the extreme left and right nodes for each level. It's a robust problem that highlights your analytical skills and grasp of tree data structures within a binary tree and BFS/DFS interview pattern.

Algorithmic pattern used

The "Maximum Width of Binary Tree" problem can be efficiently solved using Breadth-First Search (BFS). The key idea is to assign a unique position or index to each node as if it were part of a complete binary tree. If the root is at index p, its left child is at 2*p and its right child is at 2*p + 1.

1. BFS Traversal: Use a queue to perform a level-order traversal. For each level:
   • Store (node, position) pairs in the queue.
   • When processing a level, keep track of the leftmost_position and rightmost_position encountered.
   • The width of the current level is rightmost_position - leftmost_position + 1. Update the overall maximum width.
   • For the children of a node (current_node, current_pos), add (current_node.left, 2 * current_pos) and (current_node.right, 2 * current_pos + 1) to the queue if they are not null. This Breadth-First Search and binary tree interview pattern guarantees processing level by level and accurate width calculation. DFS can also be adapted by passing level and position information.

Example explanation

Consider the binary tree: 1 / 3 2 / \ 5 3 9

Using BFS with 0-indexed positions:

• Level 0:
  • Queue: [(1, 0)]
  • Process (1, 0). leftmost = 0, rightmost = 0. Current width = 0 - 0 + 1 = 1. maxWidth = 1.
  • Add children: [(3, 0*2), (2, 0*2+1)] => [(3, 0), (2, 1)] (after adjusting for children indexing starting from 0 relative to level start)
  • Let's use the absolute indexing where root is 1, left child 2p, right child 2p+1. Root 1 at index = 1.
• Level 0:
  • Queue: [(1, 1)]
  • Process (1, 1). leftmost_on_level = 1, rightmost_on_level = 1. current_width = 1 - 1 + 1 = 1. maxWidth = 1.
  • Add children: (3, 2*1), (2, 2*1+1) => [(3, 2), (2, 3)].
• Level 1:
  • Queue: [(3, 2), (2, 3)]
  • Process (3, 2). leftmost_on_level = 2. Add children: (5, 2*2), (3, 2*2+1) => (5, 4), (3, 5).
  • Process (2, 3). rightmost_on_level = 3. Add child: (9, 2*3+1) => (9, 7).
  • current_width = 3 - 2 + 1 = 2. maxWidth = max(1, 2) = 2.
  • Queue: [(5, 4), (3, 5), (9, 7)].
• Level 2:
  • Queue: [(5, 4), (3, 5), (9, 7)]
  • Process (5, 4). leftmost_on_level = 4.
  • Process (3, 5).
  • Process (9, 7). rightmost_on_level = 7.
  • current_width = 7 - 4 + 1 = 4. maxWidth = max(2, 4) = 4. The maximum width is 4. This shows how the Maximum Width of Binary Tree coding problem is solved.

Common mistakes candidates make

A common mistake in the Maximum Width of Binary Tree coding problem is defining width as merely the count of non-null nodes at a level, rather than accounting for null nodes between the leftmost and rightmost non-null nodes. Another frequent error is incorrectly assigning positions to nodes, especially when dealing with gaps or skewed trees, leading to wrong width calculations. Some might struggle with the implementation details of BFS, particularly managing the queue and identifying the start and end of each level. Overlooking potential integer overflow issues if the tree is very wide and position indices become large is also a pitfall, requiring careful choice of data types.

Interview preparation tip

To excel in the Maximum Width of Binary Tree interview question, thoroughly understand Breadth-First Search (BFS) and how to apply it level-by-level. Practice assigning conceptual indices to binary tree nodes as if they were in a complete tree array. Pay close attention to how you track the leftmost and rightmost node's indices for each level. Work through various tree examples, including skewed trees and trees with many null children, to ensure your indexing logic is robust. This binary tree, BFS, and tree traversal interview pattern is fundamental for many tree-related problems.