Classic tree DP problem but I always second-guess myself on whether the path has to pass through the root or not.
Use a recursive post-order traversal to compute the maximum path sum. At each node, calculate the maximum gain from its left and right subtrees, update the global maximum path sum considering the node as the highest point, and return the maximum gain the node can contribute to its parent.
Pro tip: Clarify that the path can start and end at any node, and handle negative values by ignoring negative gains (using max(0, gain)). This shows attention to edge cases and robustness.
Confirm that the path can start and end at any node, may not pass through the root, and that nodes can have negative values. Ask if a single node is considered a valid path.
Define a helper function that returns the maximum gain from a subtree rooted at a given node, where the gain is the maximum sum of a path starting at that node and going down to any node in its subtree.
For each node, recursively compute the left and right gains, ignoring negative gains (use max(0, gain)). Update the global maximum path sum with the sum of the node's value plus both gains.
Return the node's value plus the maximum of its left and right gains (since a path cannot split when going up). This represents the best gain the node can contribute to its parent.
State that the time complexity is O(n) and space complexity is O(h) due to recursion. Discuss edge cases like a single node, all negative values, and skewed trees.
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