Squirrel Migration

AtCoder

IDarc087_d

Time5000ms

Memory256MB

Difficulty—

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: $N$ $x_1$ $y_1$ $x_2$ $y_2$ $:$ $x_{N - 1}$ $y_{N - 1}$ [samples]

Samples

Input #1

3
1 2
2 3

Output #1

3

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

*   $(2, 3, 1)$
*   $(3, 1, 2)$
*   $(3, 2, 1)$

Input #2

Output #2

11

The maximum possible distance traveled by squirrels is $6$. For example, $p = (2, 1, 4, 3)$ achieves it.

Input #3

Output #3

Input #4

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Squirrel Migration",
    "description": {
      "content": "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 $w$ ($1 \\leq v, w \\leq N$),",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 5000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc087_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "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 $w$ ($1 \\leq v, w \\leq N$),...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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