Trees Are Fun

You are given a rooted tree consisting of $N$ nodes numbered from $1$ to $N$, where the $i^{th}$ node has color $C_i$. The root of the tree is node $1$. Let $F_i$ denote the frequency of color $i$ in the tree, and, $S_u$ denote the set of colors of all vertices in the subtree of node $u$. For each node $u$ $(1\le u \le N)$, find $\prod_{i \in S_1 \setminus S_u} F_i$. In other words, for e