Sum on Tree

You are given a tree with $N$ nodes (numbered $1$ through $N$) and $N-1$ edges. Each node has a value; let's denote the value of node $x$ by $W_x$. Next, let's define the value of a simple path $v_1, v_2, \dots, v_k$ as $\sum_{i=1}^k i \cdot W_{v_i}$. A simple path in a tree is a sequence of nodes $v_1, v_2, \dots, v_k$ such that: - $k \ge 1$ - there is an edge between nodes $v_i$ and $v_{i+1