A16082. Redundant Routes

You are given a tree with $n$ vertices labeled $1, 2, \ldots, n$ . The length of a simple path in the tree is the number of vertices in it.

You are to select a set of simple paths of length at least $2$ each, but you cannot simultaneously select two distinct paths contained one in another. Find the largest possible size of such a set.

Formally, a set $S$ of vertices is called a route if it contains at least two vertices and coincides with the set of vertices of a simple path in the tree. A collection of distinct routes is called a timetable. A route $S$ in a timetable $T$ is called redundant if there is a different route $S' \in T$ such that $S \subset S'$ . A timetable is called efficient if it contains no redundant routes. Find the largest possible number of routes in an efficient timetable.

The first line contains a single integer $n$ ( $2 \le n \le 3000$ ).

The $i$ -th of the following $n - 1$ lines contains two integers $u_i$ and $v_i$ ( $1 \le u_i, v_i \le n$ , $u_i \neq v_i$ ) — the numbers of vertices connected by the $i$ -th edge.

It is guaranteed that the given edges form a tree.

Print a single integer — the answer to the problem.

In the first example, possible efficient timetables are $\{\{1, 2\}, \{1, 3\}, \{1, 4\}\}$ and $\{\{1, 2, 3\}, \{1, 2, 4\}, \{1, 3, 4\}\}$ .

In the second example, we can choose $\{ \{1, 2, 3\}, \{2, 3, 4\}, \{3, 4, 6\}, \{2, 3, 5\}, \{3, 4, 5\}, \{3, 4, 7\}, \{4, 6, 7\}\}$ .