Score : $800$ points
There is a tree with $N$ vertices. The vertices are numbered $1$ through $N$. The $i$-th edge ($1 \leq i \leq N - 1$) connects Vertex $x_i$ and $y_i$. For vertices $v$ and $w$ ($1 \leq v, w \leq N$), we will define the distance between $v$ and $w$ $d(v, w)$ as "the number of edges contained in the path $v$-$w$".
A squirrel lives in each vertex of the tree. They are planning to move, as follows. First, they will freely choose a permutation of $(1, 2, ..., N)$, $p = (p_1, p_2, ..., p_N)$. Then, for each $1 \leq i \leq N$, the squirrel that lived in Vertex $i$ will move to Vertex $p_i$.
Since they like long travels, they have decided to maximize the total distance they traveled during the process. That is, they will choose $p$ so that $d(1, p_1) + d(2, p_2) + ... + d(N, p_N)$ will be maximized. How many such ways are there to choose $p$, modulo $10^9 + 7$?
Input is given from Standard Input in the following format:
$N$
$x_1$ $y_1$
$x_2$ $y_2$
$:$
$x_{N - 1}$ $y_{N - 1}$
Print the number of the ways to choose $p$ so that the condition is satisfied, modulo $10^9 + 7$.
3 1 2 2 3
3
The maximum possible distance traveled by squirrels is $4$. There are three choices of $p$ that achieve it, as follows:
4 1 2 1 3 1 4
11
The maximum possible distance traveled by squirrels is $6$. For example, $p = (2, 1, 4, 3)$ achieves it.
6 1 2 1 3 1 4 2 5 2 6
36
7 1 2 6 3 4 5 1 7 1 5 2 3
396