I recognized it as a tree DP problem pretty fast, which felt good, but then I fumbled around with the recursion signature for longer than I should have.
Recognize this as a tree DP problem where each node returns two values: the maximum sum when the node is selected (including its value plus the sums of its children's unselected states) and when it is not selected (sum of the maximum of its children's selected/unselected states). Use post-order DFS to compute these values bottom-up, and the answer is the maximum of the two values at the root.
Pro tip: Clarify edge cases upfront (e.g., empty tree, single node, negative values) and discuss time/space complexity (O(N) time, O(H) space for recursion) to demonstrate thoroughness. Mention that the DP can be implemented iteratively to avoid recursion depth issues for deep trees.
Confirm that the tree is N-ary, values are non-negative, and the goal is to maximize the sum with no two selected nodes adjacent. Ask about input size to gauge if recursion depth is a concern.
For each node, define two states: dp[node][0] = max sum in subtree when node is not selected, dp[node][1] = max sum when node is selected. Explain the recurrence relations.
Use post-order traversal (DFS) to compute dp values. For a leaf, dp[leaf][0] = 0, dp[leaf][1] = value. For internal nodes, combine children's results as per recurrence.
State that time complexity is O(N) since each node is visited once, and space is O(H) for recursion stack (or O(N) worst-case). Discuss handling of empty tree and single node.
Walk through a small example (e.g., root with two children) to verify the DP transitions and ensure the answer is correct. Mention potential optimizations like iterative DFS if needed.
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