Contest Duration: - (local time) (100 minutes) Back to Home
F - Squirrel Migration /

Time Limit: 5 sec / Memory Limit: 512 MB

### 問題文

N 頂点の木があります。 頂点には 1 から N まで番号が振られています。 i (1 \leq i \leq N - 1) 番目の辺は頂点 x_iy_i を結んでいます。 頂点 v, w (1 \leq v, w \leq N) について、v, w 間の距離 d(v, w) を「パス v-w に含まれる辺の本数」と定義します。

リスたちは移動が好きなので、各リスの移動距離の総和が最大になるように引っ越しをすることにしました。 すなわち、d(1, p_1) + d(2, p_2) + ... + d(N, p_N) が最大になるように p を選ぶことにしました。 このような p の選び方は何通りあるでしょうか？ 10^9 + 7 で割った余りを求めてください。

### 制約

• 2 \leq N \leq 5,000
• 1 \leq x_i, y_i \leq N
• 入力のグラフは木である。

### 入力

N
x_1 y_1
x_2 y_2
:
x_{N - 1} y_{N - 1}

3
1 2
2 3

3

• (2, 3, 1)
• (3, 1, 2)
• (3, 2, 1)

4
1 2
1 3
1 4

11

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

### 出力例 4

396

Score : 800 points

### Problem Statement

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?

### Constraints

• 2 \leq N \leq 5,000
• 1 \leq x_i, y_i \leq N
• The input graph is a tree.

### Input

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}

### Output

Print the number of the ways to choose p so that the condition is satisfied, modulo 10^9 + 7.

3
1 2
2 3

### Sample Output 1

3

The maximum possible distance traveled by squirrels is 4. There are three choices of p that achieve it, as follows:

• (2, 3, 1)
• (3, 1, 2)
• (3, 2, 1)

4
1 2
1 3
1 4

### Sample Output 2

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