Zigzag Tree

AtCoder

IDarc130_d

Time2000ms

Memory256MB

Difficulty—

Given is a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\-th edge connects Vertices $a_i$ and $b_i$. Find the number of sequences of positive integers $P = (P_1, P_2, \ldots, P_N)$ that satisfy the conditions below, modulo $998244353$. * $1\leq P_i\leq N$ * $P_i\neq P_j$ if $i\neq j$. * For $1\leq a, b, c\leq N$, if Vertices $a$ and $b$ are adjacent, and Vertices $b$ and $c$ are also adjacent, $P_a < P_b > P_c$ or $P_a > P_b < P_c$ holds. ## Constraints * $2\leq N\leq 3000$ * $1\leq a_i, b_i\leq N$ * The given graph is a tree. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $b_1$ $a_2$ $b_2$ $\vdots$ $a_{N-1}$ $b_{N-1}$ [samples]

Samples

Input #1

3
1 2
2 3

Output #1

4

The $4$ sequences satisfying the conditions are:

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

Input #2

Output #2

Input #3

Output #3

Input #4

Output #4

API Response (JSON)

{
  "problem": {
    "name": "Zigzag Tree",
    "description": {
      "content": "Given is a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects Vertices $a_i$ and $b_i$. Find the number of sequences of positive integers $P = (P_1, P_2, \\ldot",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc130_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is a tree with $N$ vertices. The vertices are numbered $1$ to $N$, and the $i$\\-th edge connects Vertices $a_i$ and $b_i$.\nFind the number of sequences of positive integers $P = (P_1, P_2, \\ldot...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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