Chef Cuts Tree

A forest is an undirected acyclic graph. Let us define the strength of a forest as the sum of squares of sizes of its connected components. (Clearly, a tree with n nodes has strength n2.) Chef has found a tree with N nodes on day 0. On each of the next N-1 days, he's going to remove one edge. Let's denote the forest that remains after i days by Fi, for each 1 ≤ i ≤ N-1; also, let's denote the or