So v is a vertex and u is one of his sons.

First notice that if x is a node in subtree of u then -> dis(u,x) = dis(v,x)-1

Then notice that in array d[v] we included u's subtree too.

So the number of nodes that are in subtree of v but not in subtree of u and have distance k from v are d[v][k]-d[u][k-1].

According to these, when we are calculating the sigma , we must calc nodes with distance x-1 from subtree of u and k-x from subtree of v but exclude those are in the subtree of u (from second one).