Fixed-Length Paths I - Implementation

Here is the solution:

function count_paths(node, adj, k)
    c := find_centroid(node, adj)
    result := count_through_centroid(c, adj, k)
    removed[c] := true
    for child in adj[c]
        if not removed[child] then
            result := result + count_paths(child, adj, k)
            return result

you will need to subtract same-subtree paths as well.

This runs in $O(n \log n)$ time and uses $O(n)$ space.