Think First - Fixed-Length Paths I

You need to count paths with a specific length k. This is not a subtree query (not asking "all nodes under v"). It is not a path value query (not asking "sum from u to v"). It is a counting problem. Naive approach: for every pair of nodes, compute the distance using LCA and check if it equals k. That is $O(n^2)$ pairs times $O(\log n)$ per LCA, way too slow for $200$,$000$ nodes. Can you split the tree into smaller pieces and count paths within each piece?

Think about recursive decomposition. What property would the split point need?