← Two Sigma Interview Insights
Classic tree DP once you recognize what it is, but I blanked for a bit on the recurrence.
Use dynamic programming on the tree with two states per node: include the node in the independent set or exclude it. Perform a post-order DFS to compute the maximum independent set size for each subtree, combining children's results. Then discuss edge cases and test design.
Pro tip: Emphasize that the DP is optimal and runs in O(n) because each node is visited once; also mention that you can reconstruct the actual set if needed, and that your tests cover structural extremes like single node, path, and star to validate both correctness and performance.
For each node u, let dp_in[u] be the max independent set size in u's subtree when u is included, and dp_out[u] when u is excluded. Recurrence: dp_in[u] = 1 + sum(dp_out[v]) for children v; dp_out[u] = sum(max(dp_in[v], dp_out[v])).
Use iterative post-order DFS (or recursion with increased recursion limit) to process children before parent. Compute dp_in and dp_out bottom-up, then answer is max(dp_in[root], dp_out[root]).
Each node and edge is processed once, so time is O(n) and space is O(n) for the DP arrays and recursion/stack. Mention that this is optimal for tree DP.
Create tests for: single node (answer 1), path graph of n nodes (answer ceil(n/2)), star graph with center and k leaves (answer max(1, k) = k for k>=1), and a balanced binary tree. Also test n=0 if allowed.
Run tests, check for off-by-one and recursion depth issues. Optionally discuss reconstructing the set by storing choices, and note that the same DP works for weighted independent set.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.