Distributing Integers

AtCoder

IDabc160_f

Time3000ms

Memory256MB

Difficulty—

We have a tree with $N$ vertices numbered $1$ to $N$. The $i$\-th edge in this tree connects Vertex $a_i$ and $b_i$. For each $k=1, ..., N$, solve the problem below: * Consider writing a number on each vertex in the tree in the following manner: * First, write $1$ on Vertex $k$. * Then, for each of the numbers $2, ..., N$ in this order, write the number on the vertex chosen as follows: * Choose a vertex that still does not have a number written on it and is adjacent to a vertex with a number already written on it. If there are multiple such vertices, choose one of them at random. * Find the number of ways in which we can write the numbers on the vertices, modulo $(10^9+7)$. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq a_i,b_i \leq N$ * The given graph is a tree. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $b_1$ $:$ $a_{N-1}$ $b_{N-1}$ [samples]

Samples

Input #1

3
1 2
1 3

Output #1

2
1
1

The graph in this input is as follows:
![image](https://img.atcoder.jp/ghi/1c88b0eb716ba399b1c5d6565ab62337.png)
For $k=1$, there are two ways in which we can write the numbers on the vertices, as follows:

*   Writing $1, 2, 3$ on Vertex $1, 2, 3$, respectively
*   Writing $1, 3, 2$ on Vertex $1, 2, 3$, respectively

Input #2

2
1 2

Output #2

1
1

The graph in this input is as follows:
![image](https://img.atcoder.jp/ghi/c47c7798f88e7bfec30fbd664dc9ad50.png)

Input #3

Output #3

2
8
12
3
3

The graph in this input is as follows:
![image](https://img.atcoder.jp/ghi/e9c09403f8d96ae4e679a226993defa6.png)

Input #4

Output #4

40
280
840
120
120
504
72
72

The graph in this input is as follows:
![image](https://img.atcoder.jp/ghi/a85459a03d436560bfe2e911d8cec4e6.png)

API Response (JSON)

{
  "problem": {
    "name": "Distributing Integers",
    "description": {
      "content": "We have a tree with $N$ vertices numbered $1$ to $N$. The $i$\\-th edge in this tree connects Vertex $a_i$ and $b_i$. For each $k=1, ..., N$, solve the problem below: *   Consider writing a number on ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 3000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "abc160_f"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a tree with $N$ vertices numbered $1$ to $N$. The $i$\\-th edge in this tree connects Vertex $a_i$ and $b_i$. For each $k=1, ..., N$, solve the problem below:\n\n*   Consider writing a number on ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments