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++14 (GCC 5.4.1) (judge verdict: Wrong Answer)

#include <bits/stdc++.h>
#define pb push_back
#define int long long
#define mp make_pair
#define inf 1000000007
#define ll long long
using namespace std;
vector<int> nex[114514];
int deg[114514],used[114514];
signed main(){
	int n;
	cin>>n;
	for(int i=0;i<n-1;i++){
		int x,y;
		cin>>x>>y;
		nex[x].pb(y);
		nex[y].pb(x);
		deg[x]++;
		deg[y]++;
	}
	queue<int> que;
	for(int i=1;i<=n;i++){
		if(deg[i]==1){
			que.push(deg[i]);
		}
	}
	while(!que.empty()){
		int cur = que.front(); que.pop();
		if(used[cur]==0){
			for(int i=0;i<nex[cur].size();i++){
				int dur = nex[cur][i];
				if(used[dur]==0){
					used[cur]=dur;
					used[dur]=cur;
					deg[cur]=0;
					deg[dur]=0;
					for(int j=0;j<nex[dur].size();j++){
						if( deg [ nex[dur][j] ] >= 1){
							deg[ nex[dur][j] ]--;
						}
						if( deg[nex[dur][j]] == 1){
							que.push(nex[dur][j]);
						}
					}
				}
			}
		}
	}
	for(int i=1;i<=n;i++){
		if(used[i]==0){
			cout<<"First"<<endl;
			return 0;
		}
	}
	cout<<"Second"<<endl;
	return 0;
}