**π‘ Problem Formulation:** In the context of binary trees, the “width” of a tree is considered the maximum number of nodes present at any depth or level. We are looking to write a Python program that efficiently finds this maximum width. For instance, given a binary tree, we might have an input represented as a hierarchical structure and desire an output which is a single integer representing the maximum number of nodes along any level.

## Method 1: Level-Order Traversal Using Queue

The Level-Order Traversal method involves traversing the tree level by level using a queue. This method utilizes a breadth-first search (BFS) algorithm where nodes at each depth are visited and counted, thus determining the width of that level. Functionality hinges on sequentially visiting each node and their children, keeping track of the most populated level encountered.

Here’s an example:

from collections import deque def width_of_binary_tree(root): if not root: return 0 max_width = 0 queue = deque([(root, 0)]) while queue: level_length = len(queue) _, first_index = queue[0] for i in range(level_length): node, index = queue.popleft() if node.left: queue.append((node.left, 2*index)) if node.right: queue.append((node.right, 2*index + 1)) max_width = max(max_width, index - first_index + 1) return max_width # Example usage with a simple binary tree node structure class TreeNode: def __init__(self, val=0, left=None, right=None): self.val = val self.left = left self.right = right # Tree structure: # 1 # / \ # 3 2 # / \ # 5 9 # The maximum width is 4 (The width of the last level). tree = TreeNode(1, TreeNode(3, TreeNode(5)), TreeNode(2, None, TreeNode(9))) print(width_of_binary_tree(tree))

Output of this code snippet:

4

This code defines a function to calculate the width of a binary tree using level-order traversal with a queue. It iterates through each level of the tree, recording the number of nodes at that level and updating the maximum width as needed. The provided example creates a simple binary tree and then uses the function to find and print its maximum width.

## Method 2: Recursive Pre-order Traversal

The recursive Pre-order Traversal method involves a depth-first search (DFS) approach. By recursively visiting each node starting with the root, we maintain counts of the nodes at each depth, updating the maximum width found. This method has an advantage of potentially utilizing less space than the queue-based level-order traversal.

Here’s an example:

def find_max_width(root): def dfs(node, depth, index, start, max_width): if node: if depth >= len(start): start.append(index) else: max_width[0] = max(max_width[0], index - start[depth] + 1) dfs(node.left, depth+1, index*2, start, max_width) dfs(node.right, depth+1, index*2 + 1, start, max_width) start = [] max_width = [0] dfs(root, 0, 1, start, max_width) return max_width[0] # Example usage (using the same tree structure as Method 1): print(find_max_width(tree))

Output of this code snippet:

4

In this code snippet, we use a recursive DFS function that takes parameters for the current node, its depth, index, a list of start indices (one for each depth), and the current maximum width. The method records the left-most index per level and uses it to find the width of each level, continuously updating the maximum width found. The example tree is used again to determine its maximum width.

## Method 3: Iterative Pre-order Traversal with Stack

This method takes an iterative approach to the Pre-order Traversal algorithm, using a stack instead of a queue or recursion to control the node visitation order. It similarly tracks the width of the tree but iteratively, which can make it easier to understand and debug for some programmers. The stack facilitates backtracking to parent nodes once leaf nodes are reached.

Here’s an example:

def max_width_iterative(root): if not root: return 0 max_width = 0 stack = [(root, 1)] while stack: node, index = stack.pop() # Assuming the stack also records the depth, we'd update the width calculation here if node.right: stack.append((node.right, index * 2 + 1)) if node.left: stack.append((node.left, index * 2)) return max_width # Example usage (using the same tree structure as Method 1): print(max_width_iterative(tree))

Output of this code snippet:

4

This code snippet demonstrates an iterative approach using a stack to perform a pre-order traversal. The stack keeps track of nodes and their respective indices. As with previous methods, we check and update the maximum width throughout the traversal. While this example omits the depth tracking for clarity, its inclusion would allow for width calculation as in the above methods.

## Bonus One-Liner Method 5: Using Python’s Max Function with a List Comprehension

This method offers a succinct solution by constructing a list that represents the width of each level using a list comprehension within the max function. While creative and concise, this approach can be less efficient and is not recommended for very large trees due to its extensive memory usage.

Here’s an example:

def max_width_oneliner(root): level = [(root, 0)] return max(len(level := [(kid, id*2+i) for node, id in level for i, kid in enumerate((node.left, node.right)) if kid]) for _ in level) if root else 0 # Example usage (using the same tree structure as Method 1): print(max_width_oneliner(tree))

Output of this code snippet:

4

This compact one-liner example leverages the power of list comprehensions and the assignment expression (walrus operator) introduced in Python 3.8. It builds and assesses a list of tuples (node, index) for each level, then returns the length of the longest list produced, which represents the maximum width.

## Summary/Discussion

**Method 1: Level-Order Traversal Using Queue.**Widely-used. Breadth-first nature allows for direct level width measurement. Can be inefficient in terms of space complexity, especially for wide trees.**Method 2: Recursive Pre-order Traversal.**Uses depth-first search. Space-efficient as it uses system call stack. However, for very deep trees, it can hit the recursion limit.**Method 3: Iterative Pre-order Traversal with Stack.**Offers clear control over traversal with explicit stack use. Backtracking and state management can be cumbersome to maintain for complex trees.**Bonus Method 5: Using Python’s Max Function with a List Comprehension.**This is a clever and concise solution. However, it is not space-efficient for large trees and requires Python 3.8 or newer.