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 pairs times per LCA, way too slow for , 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?