If you fix son u of vertex v and distance x to the v, you can choose vertex with distance to the v k — x in subtree of other son of v. d[u][x — 1] is number of vertices in subtree of u with distance x — 1(becomes x when you move from u to v). d[v][k — x] is number of vertices in subtree of v with distance k — x, you subtract d[u][k — x — 1] to exclude vertices in subtree of u. Also *0.5 is because each pair is counted twice.