Black and White Tree

AtCoder

IDagc014_d

Time2000ms

Memory256MB

Difficulty—

There is a tree with $N$ vertices numbered $1$ through $N$. The $i$\-th of the $N-1$ edges connects vertices $a_i$ and $b_i$. Initially, each vertex is uncolored. Takahashi and Aoki is playing a game by painting the vertices. In this game, they alternately perform the following operation, starting from Takahashi: * Select a vertex that is not painted yet. * If it is Takahashi who is performing this operation, paint the vertex white; paint it black if it is Aoki. Then, after all the vertices are colored, the following procedure takes place: * Repaint every white vertex that is adjacent to a black vertex, in black. Note that all such white vertices are repainted simultaneously, not one at a time. If there are still one or more white vertices remaining, Takahashi wins; if all the vertices are now black, Aoki wins. Determine the winner of the game, assuming that both persons play optimally. ## Constraints * $2 ≤ N ≤ 10^5$ * $1 ≤ a_i,b_i ≤ N$ * $a_i ≠ b_i$ * The input 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
2 3

Output #1

First

Below is a possible progress of the game:

*   First, Takahashi paint vertex $2$ white.
*   Then, Aoki paint vertex $1$ black.
*   Lastly, Takahashi paint vertex $3$ white.

In this case, the colors of vertices $1$, $2$ and $3$ after the final procedure are black, black and white, resulting in Takahashi's victory.

Input #2

Output #2

First

Input #3

Output #3

Second

API Response (JSON)

{
  "problem": {
    "name": "Black and White Tree",
    "description": {
      "content": "There is a tree with $N$ vertices numbered $1$ through $N$. The $i$\\-th of the $N-1$ edges connects vertices $a_i$ and $b_i$. Initially, each vertex is uncolored. Takahashi and Aoki is playing a game ",
      "description_type": "Markdown"
    },
    "platform": "AtCoder",
    "limit": {
      "time_limit": 2000,
      "memory_limit": 262144
    },
    "difficulty": "None",
    "is_remote": true,
    "is_sync": true,
    "sync_url": null,
    "sign": "agc014_d"
  },
  "statements": [
    {
      "statement_type": "Markdown",
      "content": "There is a tree with $N$ vertices numbered $1$ through $N$. The $i$\\-th of the $N-1$ edges connects vertices $a_i$ and $b_i$.\nInitially, each vertex is uncolored.\nTakahashi and Aoki is playing a game ...",
      "is_translate": false,
      "language": "English"
    }
  ]
}

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