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>
#include<ext/pb_ds/assoc_container.hpp>
#include<ext/pb_ds/tree_policy.hpp>
#define pi 3.141592653589793238
#define int long long
using namespace __gnu_pbds;
using namespace std;
template <typename T>
using ordered_set = tree<T, null_type, less<T>, rb_tree_tag, tree_order_statistics_node_update>;

vector<vector<int> >adj;
int mm[100005];
int cnt[100005];
int mm2[100005];

signed main()
{
ios_base::sync_with_stdio(false);
cin.tie(NULL);  
cout.tie(0);   

#ifndef ONLINE_JUDGE
	if(fopen("INPUT.txt","r"))
	{
	freopen ("INPUT.txt" , "r" , stdin);
	freopen ("OUTPUT.txt" , "w" , stdout);
	}
#endif  

	int n;
	cin>>n;
	adj.resize(n+1);
	int x,y;
	for(int i=0;i<n-1;i++)
	{
		cin>>x>>y;
		adj[x].push_back(y);
		adj[y].push_back(x);		
	}
	for(int i=1;i<=n;i++)
	{
		if(adj[i].size()==1)
			mm[i]=1;
	}

	for(int i=1;i<=n;i++)
	{
		for(auto j:adj[i])
		{
			cnt[i]+=mm[j];
		}
		if(cnt[i]>1)
		{
			cout<<"First";
			return 0;
		}
	}
	for(int i=1;i<=n;i++)
	{
		int count=0;
		if(cnt[i]||mm[i])
			continue;
		for(auto j:adj[i])
		{
			count+=cnt[j];
		}
		if(count>1 && adj[i].size()==count)
		{
			cout<<"First";
			return 0;
		}
		mm2[i]=count;
	}

	for(int i=1;i<=n;i++)
	{
		int count=0;
		if(mm2[i]||mm[i]||cnt[i])
			continue;
		for(auto j:adj[i])
		{
			count+=mm2[j];
		}
		if(count>1 && adj[i].size()==count)
		{
			cout<<"First";
			return 0;
		}
	}


	cout<<"Second";

}