Problem:
Score : 900 points

Problem Statement

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}

Output

Print First if Takahashi wins; print Second if Aoki wins.

Sample Input 1

3
1 2
2 3

Sample 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.

Sample Input 2

4
1 2
2 3
2 4

Sample Output 2

First

Sample Input 3

6
1 2
2 3
3 4
2 5
5 6

Sample Output 3

Second

Solution in C# (Mono 4.6.2.0) (judge verdict: Accepted)

using System;
using System.Collections.Generic;

public class CSharp{
    public static void Main(){
        Kyopuro chokudai = new Kyopuro();
	    chokudai.AtCoder();
	}
}

public class Kyopuro{
    int n;
    List<List<int>> list;
    List<int>[] tree;
    List<int>[] ptree;
    bool[] v;
    int[] star;
    bool ans;
    
    public void AtCoder(){
        n = int.Parse(Console.ReadLine());
        tree = new List<int>[n];
        ptree = new List<int>[n];
        for (int i=0; i<n; i++){
            tree[i] = new List<int>();
            ptree[i] = new List<int>();
        }
        string[] s;
        for (int i = 0; i < n - 1; i++){
		    s = Console.ReadLine().Split(' ');
		    int a = int.Parse(s[0]) - 1;
		    int b = int.Parse(s[1]) - 1;
		    tree[a].Add(b);
		    tree[b].Add(a);
        }
        
        ans = false;
        v = new bool[n];
        v[0] = true;
        star = new int[n];
        list = new List<List<int>>();
        list.Add(new List<int>());
        list[0].Add(0);
        Bfs(1);
        int cc = 0;
        foreach(int x in ptree[0]){
            if(star[x] == 1) cc++;
        }
        if(cc != 1) ans = true;
        if(ans) Console.WriteLine("First");
        else Console.WriteLine("Second");
    }
    
    void Bfs(int d){
        list.Add(new List<int>());
        foreach(int p in list[d-1]){
            foreach(int q in tree[p]){
                if(!v[q]){
                    list[d].Add(q);
                    ptree[p].Add(q);
                    v[q] = true;
                }
            }
        }
        if(list[d].Count > 0){
            Bfs(d+1);
            foreach(int x in list[d]){
                star[x] = 1;
                int cc = 0;
                foreach(int y in ptree[x]){
                    if(star[y] == 1){
                        cc++;
                        star[x] = 0;
                    }
                }
                if(cc>1) ans = true;
            }
        }
    }
    
    public static void DebugWL(object x){ System.Diagnostics.Debug.WriteLine(x); }
}