A13872. Range Diameter Sum

You are given a tree with $n$ vertices numbered $1, \ldots, n$ . A tree is a connected simple graph without cycles.

Let $\mathrm{dist}(u, v)$ be the number of edges in the unique simple path connecting vertices $u$ and $v$ .

Let $\mathrm{diam}(l, r) = \max \mathrm{dist}(u, v)$ over all pairs $u, v$ such that $l \leq u, v \leq r$ .

Compute $\sum_{1 \leq l \leq r \leq n} \mathrm{diam}(l, r)$ .

The first line contains a single integer $n$ ( $1 \leq n \leq 10^5$ ) — the number of vertices in the tree.

The next $n - 1$ lines describe the tree edges. Each of these lines contains two integers $u, v$ ( $1 \leq u, v \leq n$ ) — endpoint indices of the respective tree edge. It is guaranteed that the edge list indeed describes a tree.

Print a single integer — $\sum_{1 \leq l \leq r \leq n} \mathrm{diam}(l, r)$ .