Redundant Routes
Description
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.
Input
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.
Output
Print a single integer — the answer to the problem.
4
1 2
1 3
1 4
7
2 1
3 2
4 3
5 3
6 4
7 4
3
7
Note
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 $\{ \{1, 2, 3\}, \{2, 3, 4\}, \{3, 4, 6\}, \{2, 3, 5\}, \{3, 4, 5\}, \{3, 4, 7\}, \{4, 6, 7\}\}$.