Swap on Tree

AtCoder

IDarc171_c

Time2000ms

Memory256MB

Difficulty—

There is a tree with $N$ vertices numbered $1$ to $N$. The $i$\-th edge connects vertices $u_i$ and $v_i$. Additionally, there are $N$ pieces numbered $1$ to $N$. Initially, piece $i$ is placed on vertex $i$. You can perform the following operation any number of times, possibly zero: * Choose one edge. Let vertices $u$ and $v$ be the endpoints of the edge, and swap the pieces on vertices $u$ and $v$. Then, delete the chosen edge. Let $a_i$ be the piece on vertex $i$. How many different possible sequences $(a_1, a_2, \dots, a_N)$ exist when you finish performing the operation? Find the count modulo $998244353$. ## Constraints * $2 \leq N \leq 3000$ * $1 \leq u_i \lt v_i \leq N$ * The graph given in the input is a tree. ## Input The input is given from Standard Input in the following format: $N$ $u_1$ $v_1$ $u_2$ $v_2$ $\vdots$ $u_{N-1}$ $v_{N-1}$ [samples]

Samples

Input #1

3
1 2
2 3

Output #1

5

For example, the sequence $(a_1, a_2, a_3) = (2, 1, 3)$ can be obtained by the following steps:

*   Choose the first edge, swap the pieces on vertices $1$ and $2$, and delete the edge. This results in $(a_1, a_2, a_3) = (2, 1, 3)$.
*   Finish operating.

Also, the sequence $(a_1, a_2, a_3) = (3, 1, 2)$ can be obtained by the following steps:

*   Choose the second edge, swap the pieces on vertices $2$ and $3$, and delete the edge. This results in $(a_1, a_2, a_3) = (1, 3, 2)$.
*   Choose the first edge, swap the pieces on vertices $1$ and $2$, and delete the edge. This results in $(a_1, a_2, a_3) = (3, 1, 2)$.
*   Finish operating.

The operation can yield the following five sequences:

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

Input #2

Output #2

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Swap on Tree",
    "description": {
      "content": "There is a tree with $N$ vertices numbered $1$ to $N$. The $i$\\-th edge connects vertices $u_i$ and $v_i$.   Additionally, there are $N$ pieces numbered $1$ to $N$. Initially, piece $i$ is placed on v",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc171_c"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a tree with $N$ vertices numbered $1$ to $N$. The $i$\\-th edge connects vertices $u_i$ and $v_i$.  \nAdditionally, there are $N$ pieces numbered $1$ to $N$. Initially, piece $i$ is placed on v...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments