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: Wrong Answer)

using System;
using System.Text;
using System.Collections.Generic;
using System.Linq;
class Solve{
    List<int>[] G;
    bool b;
    bool[] us;
    public Solve(){}
    StringBuilder sb;
    public static int Main(){
        new Solve().Run();
        return 0;
    }
    void Run(){
        sb = new StringBuilder();
        Calc();
        Console.Write(sb.ToString());
    }
    void Calc(){
        int N = int.Parse(Console.ReadLine());
        if(N == 2){
            sb.Append("Second\n");
            return;
        }
        G = new List<int>[N];
        for(int i=0;i<N;i++){
            G[i] = new List<int>();
        }
        for(int i=0;i<N-1;i++){
            string[] str = Console.ReadLine().Split(' ');
            int a = int.Parse(str[0]) - 1;
            int b = int.Parse(str[1]) - 1;
            G[a].Add(b);
            G[b].Add(a);
        }
        us = new bool[N];
        if(G[0].Count == 1){
            if(dfs(G[0][0]) == 0){
                b = true;
            }
        }
        else{
            if(dfs(0) == 0){
                b = true;
            }
        }
        sb.Append((b ? "First" : "Second")+"\n");
    }
    int dfs(int v){
        us[v] = true;
        if(G[v].Count == 1){
            return 1;
        }
        int state = 0;
        for(int i=0;i<G[v].Count;i++){
            int t = G[v][i];
            if(!us[t]){
                int j = dfs(t);
                if(state == 0){
                    if(j == 1){
                        state = 1;
                    }
                }
                else{
                    if(j == 1){
                        b = true;
                    }
                }
            }
        }
        if(state == 1){
            return 0;
        }
        else{
            return 1;
        }
    }
}