K-Tuple Tree

Chef opened a chain of restaurants in Chefland and invited his $K$ friends (numbered $1$ through $K$) to move there. Chefland can be represented by a tree with $N$ vertices (numbered $1$ through $N$). For each valid $i$, the $i$-th vertex has a type $c_i$. There are $K+1$ types of vertices: $c_i = 0$ means that there is a restaurant in vertex $i$, while $1 \le c_i \le K$ means that there is a h