Beauty of Pairs

You are given a tree with $N$ vertices numbered 1 through $N$ as a sequence of $N-1$ edges — pairs $(a_1, b_1), (a_2, b_2), \ldots, (a_{N-1}, b_{N-1})$. Let's define the *value* of a sequence $A$ with length $N-1$ as $$F(A)=\sum_{i=1}^{N-1} A_i \cdot W_i \,,$$ where $W_1, W_2, \ldots, W_{N-1}$ are given weights. You may change the order of the pairs and the order of elements within any pairs. For