Tree Colouring (Hard)

**This is worth $100$ points in Division $1$ and $50$ points in Divisions $2$, $3$ and $4$.** You have a tree with $N$ nodes numbered $1$ to $N$. You want to assign each node a colour $C_i$ ($1 \le C_i \le N$) such that the following conditions hold: - Each colour $X$ appears either $0$ times or $\ge 2$ times. - Consider all possible pairs $(A, B)$ such that $1 \le A < B \le N$ and $C_A = C_B$