Unfazed By All Shades

You are given a tree with $N$ vertices and $X$ available colors. Each vertex must be colored using one of the $X$ colors. Let $\text{color}(u)$ denote the color assigned to vertex $u$. Let $\mathrm{dist}(u, v)$ denote the number of edges on the unique simple path between vertices $u$ and $v$. Let $$ D = \max_{(u,v)} (\operatorname{dist}(u,v)) $$ denote the maximum distance between any two vert