Deterministic Placing

AtCoder

IDarc142_d

Time2000ms

Memory256MB

Difficulty—

We have a tree with $N$ vertices, numbered $1, \ldots, N$. For each $i=1,\ldots,N-1$, the $i$\-th edge connects Vertex $a_i$ and Vertex $b_i$. An operation on this tree when at most one piece is put on each vertex is defined as follows. * Simultaneously move every piece to one of the vertices adjacent to the one occupied by the piece. An operation is said to be **good** when the conditions below are satisfied. * Each edge is traversed by at most one piece. * Each vertex will be occupied by at most one piece after the operation. Takahashi will put a piece on one or more vertices of his choice. Among the $2^N-1$ ways to put pieces, find the number of ones that satisfy the condition below, modulo $998244353$. * For every non-negative integer $K$, the following holds. * It is possible to perform a good operation $K$ times. * Let $S_K$ be the set of vertices occupied by pieces just after $K$ good operations. Then, $S_K$ is unique. ## Constraints * $2 \leq N \leq 2 \times 10^5$ * $1 \leq a_i \lt b_i \leq N$ * The given graph is a tree. * All values in input are integers. ## Input Input is given from Standard Input in the following format: $N$ $a_1$ $b_1$ $\vdots$ $a_{N-1}$ $b_{N-1}$ [samples]

Samples

Input #1

3
1 2
1 3

Output #1

2

Here are the two ways to satisfy the condition.

*   Put a piece on Vertex $1$ and Vertex $2$.
*   Put a piece on Vertex $1$ and Vertex $3$.

Note that he must put a piece on at least one vertex.

Input #2

Output #2

0

There might be no way to satisfy the condition.

Input #3

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Deterministic Placing",
    "description": {
      "content": "We have a tree with $N$ vertices, numbered $1, \\ldots, N$. For each $i=1,\\ldots,N-1$, the $i$\\-th edge connects Vertex $a_i$ and Vertex $b_i$.   An operation on this tree when at most one piece is put",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "arc142_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "We have a tree with $N$ vertices, numbered $1, \\ldots, N$. For each $i=1,\\ldots,N-1$, the $i$\\-th edge connects Vertex $a_i$ and Vertex $b_i$.  \nAn operation on this tree when at most one piece is put...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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