Tree Sequences

You are given a tree with $N$ vertices numbered $1$ through $N$. For each valid $i$, initially, vertex $i$ has value $W_i$. The value of each vertex can be either $-1$ or a non-negative integer. At any time, let's denote the sequence of values of vertices on the path from vertex $u$ to vertex $v$ by $S(u, v)$. We are interested in replacing each element $-1$ in such a sequence by a non-negative i