# enumerative combinatorics – Calculating number of vertex-pairs with separate ancestor

Given a tree-graph with one of the vertices designated as the root, what is the complexity of calculating the number of vertex-pairs $$(u,v)$$ of which $$v$$ is not nearer to the root than $$u$$ and $$u$$ is not a vertex of the path from the root to $$v$$?

It is of course possible to determine that number in $$O(n^2)$$ by checking each pair’s Lowest Common Ancestor, but is there a way of more directly calculating that number?