Partition the Graph

You are given a tree $G$ with $N$ vertices numbered $1$ through $N$. It is guaranteed that $N$ is even. For a positive integer $k$, let's define a graph $H_k$ as follows: - $H_k$ has $N$ vertices numbered $1$ through $N$. - For each edge $(u, v)$ in $G$, there is an edge $(u, v)$ in $H_k$ too. - For each pair of vertices $(u, v)$ in $G$ such that their distance is at most $k$, there is an edge $(