**π‘ Problem Formulation:** We need to determine the maximum sum along any path between two nodes in a binary tree. The binary tree is a fundamental data structure where each node can have at most two children. The solution should be able to find, for any given binary tree, the path that has the highest sum of node values and return that sum. For instance, if the input is a tree where the highest sum path is from the root to the left child to the left grandchild with node values of 10, 9, and 7, the desired output would be 26.

## Method 1: Recursive Depth-First Search (DFS)

This method uses recursive DFS to traverse the tree and compute sums for all paths in the binary tree. The key function specification includes locating and summing the maximum path between any two nodes by backtracking from leaf nodes to the root and updating the maximum value as the recursion unwinds.

Here’s an example:

class TreeNode: def __init__(self, val=0, left=None, right=None): self.val = val self.left = left self.right = right def maxPathSum(root): def helper(node): if not node: return 0 left = max(0, helper(node.left)) right = max(0, helper(node.right)) max_path[0] = max(max_path[0], node.val + left + right) return node.val + max(left, right) max_path = [float('-inf')] helper(root) return max_path[0] # Example tree root = TreeNode(10) root.left = TreeNode(9) root.right = TreeNode(20) root.left.left = TreeNode(7) root.right.left = TreeNode(15) root.right.right = TreeNode(7) print(maxPathSum(root))

Output:

`42`

The example defines a binary tree and uses the `maxPathSum()`

function to find the largest sum of the path between two nodes. The function calculates the max path sum by considering each path through the tree only once using a helper method that traverses nodes with DFS. It keeps track of the current maximum path sum in a list `max_path`

as a workaround to the lack of non-local variable updates in Python’s nested functions.

## Method 2: Iterative DFS Using Stacks

This method follows the same DFS principles as the recursive approach but utilizes an iterative strategy with a stack to perform the traversal and compute the path sums. This is particularly useful for avoiding the overhead of recursion and potential stack overflows on very deep trees.

Here’s an example:

class TreeNode: def __init__(self, val=0, left=None, right=None): self.val = val self.left = left self.right = right def maxPathSum(root): if not root: return 0 max_path = float('-inf') stack = [(root, 0)] while stack: node, running_sum = stack.pop() current_sum = running_sum + node.val max_path = max(max_path, current_sum) if node.left: stack.append((node.left, current_sum)) if node.right: stack.append((node.right, current_sum)) return max_path # Example tree root = TreeNode(10) root.left = TreeNode(9) root.right = TreeNode(20) root.left.left = TreeNode(7) root.right.left = TreeNode(15) root.right.right = TreeNode(7) print(maxPathSum(root))

Output:

`42`

This code snippet demonstrates the iterative approach to the same problem. A stack is used to traverse the tree nodes without recursion. For each node visited, the stack stores a tuple consisting of the node and the running sum of the path so far. The `max_path`

is updated as the stack is processed.

## Method 3: Dynamic Programming on Trees

Dynamic Programming (DP) can be used on trees to optimize repeated subproblems, like re-computing the maximum path sum from a node’s children. In this method, each node’s maximum contribution to the path sum is computed once and stored for reuse.

Here’s an example:

class TreeNode: def __init__(self, val=0, left=None, right=None): self.val = val self.left = left self.right = right def maxPathSum(root): node_max_sum = {} def helper(node): if not node: return 0 # Check if node's max sum has been computed if node in node_max_sum: return node_max_sum[node] left = max(0, helper(node.left)) right = max(0, helper(node.right)) # Store the computed max sum for the node node_max_sum[node] = node.val + max(left, right) return node_max_sum[node] helper(root) return max(node_max_sum.values()) # Example tree root = TreeNode(10) root.left = TreeNode(9) root.right = TreeNode(20) root.left.left = TreeNode(7) root.right.left = TreeNode(15) root.right.right = TreeNode(7) print(maxPathSum(root))

The output and tree structure would be the same as in the previous examples. The code here features a dynamic programming approach for computing the maximum path sum, where `node_max_sum`

is a dictionary that memoizes the computed sums for each node. This avoids redundant calculations and speeds up the search for the maximum path sum.

## Method 4: Divide and Conquer

In this approach, the binary tree is essentially divided into smaller subproblems; the maximum path sum for a node includes the maximum path through its left and right subtree. This divide and conquer strategy simplifies the problem into manageable parts, and the solutions of these parts are combined to solve the overall problem.

Here’s an example:

class TreeNode: # TreeNode definition remains the same ... def maxPathSum(root): def helper(node): if not node: return (0, float('-inf')) left_single_path, left_max_path = helper(node.left) right_single_path, right_max_path = helper(node.right) single_path = node.val + max(0, left_single_path, right_single_path) max_path = max(left_max_path, right_max_path, node.val + left_single_path + right_single_path) return (single_path, max_path) return helper(root)[1] # Example tree structure remains the same ... print(maxPathSum(root))

While the output remains the same, the method here utilizes a divide and conquer strategy. The helper function returns two values for each node: the maximum single path sum and the overall maximum path sum. The single path sum is the larger of either the left or right single path sum plus the node’s value, while the overall max path sum also considers paths across the node.

## Bonus One-Liner Method 5: Tree Flattening and Max Path Calculation

This one-liner leverages Python’s list comprehensions and the itertools module to flatten the tree into a list of node values and then iterates over all possible paths to find the maximum sum. Note that while elegant, this method is not as efficient due to its brute-force nature.

Here’s an example:

import itertools # TreeNode definition remains the same ... def maxPathSum(root): def flatten(root): return flatten(root.left) + [root.val] + flatten(root.right) if root else [] return max(sum(path) for i in range(len(values := flatten(root))) for path in itertools.combinations(values, i)) # Example tree structure remains the same ... print(maxPathSum(root))

This one-liner is a conceptual illustration of using itertools and list comprehensions to compute the maximum path sum by flattening the tree and then iterating over all possible paths. Although this is a brute-force method and thus highly inefficient for large trees, it showcases the versatility of Python for compact coding techniques.

## Summary/Discussion

**Method 1:**Recursive DFS. Strengths: Simple and intuitive. Weaknesses: Can cause stack overflow in deep trees.**Method 2:**Iterative DFS Using Stacks. Strengths: Avoids recursion overhead and stack overflow issues. Weaknesses: More complex to implement and understand.**Method 3:**Dynamic Programming on Trees. Strengths: Optimizes repeated calculations, enhancing performance. Weaknesses: Requires additional space for memoization.**Method 4:**Divide and Conquer. Strengths: Simplifies complex problem into smaller parts. Weaknesses: Might be trickier to conceptualize for those new to this approach.**Bonus Method 5:**Tree Flattening and Max Path Calculation. Strengths: Elegant one-liner solution. Weaknesses: Highly inefficient for non-trivial tree sizes; more of a conceptual solution.