Counting of Trees

AtCoder

IDnikkei2019_2_qual_b

Time2000ms

Memory256MB

Difficulty—

Given is an integer sequence $D_1,...,D_N$ of $N$ elements. Find the number, modulo $998244353$, of trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition: * For every integer $i$ from $1$ to $N$, the distance between Vertex $1$ and Vertex $i$ is $D_i$. ## Constraints * $1 \leq N \leq 10^5$ * $0 \leq D_i \leq N-1$ ## Input Input is given from Standard Input in the following format: $N$ $D_1$ $D_2$ $...$ $D_N$ [samples] ## Notes * A tree of $N$ vertices is a connected undirected graph with $N$ vertices and $N-1$ edges, and the distance between two vertices are the number of edges in the shortest path between them. * Two trees are considered different if and only if there are two vertices $x$ and $y$ such that there is an edge between $x$ and $y$ in one of those trees and not in the other.

Samples

Input #1

4
0 1 1 2

Output #1

2

For example, a tree with edges $(1,2)$, $(1,3)$, and $(2,4)$ satisfies the condition.

Input #2

4
1 1 1 1

Output #2

Input #3

7
0 3 2 1 2 2 1

Output #3

API Response (JSON)

{
  "problem": {
    "name": "Counting of Trees",
    "description": {
      "content": "Given is an integer sequence $D_1,...,D_N$ of $N$ elements. Find the number, modulo $998244353$, of trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition: *   For every int",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "nikkei2019_2_qual_b"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "Given is an integer sequence $D_1,...,D_N$ of $N$ elements. Find the number, modulo $998244353$, of trees with $N$ vertices numbered $1$ to $N$ that satisfy the following condition:\n\n*   For every int...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

Full JSON Raw Segments