Lets fix centroid C of current tree, so we can find all pathes with length k thats passes throw C. How to do it: fix the order of subtrees of C and process it one by one, we should store map with pairs <dist from C, count>, and when we process subtree for each vertex with dist X from C we should add map[K-X] to answer, after subtree will be processed, we should update the map. Hope it helps.