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: Accepted)

#include <iostream>
#include <algorithm>
#include <vector>
#include <set>
#include <bitset>

using namespace std;

int n;
vector<int> graph[500000];
int col[500000];
int sz[500000];

void dfs(int v)
{
    col[v] = 1;
    sz[v] = 1;
    for (auto u : graph[v])
    {
        if (col[u] == 0)
        {
            dfs(u);
            sz[v] += sz[u];
        }
    }
}

int main()
{
    cin >> n;
    if (n % 2 == 1)
    {
        cout << "First";
        return 0;
    }
    for (int i = 0; i < n - 1; i++)
    {
        int v, u;
        cin >> v >> u;
        graph[v].push_back(u);
        graph[u].push_back(v);
    }
    dfs(1);
    for (int i = 1; i <= n; i++)
    {
        int cnt = 0;
        for (auto u : graph[i])
        {
            if (min(sz[u], sz[i]) % 2 == 1) cnt++;
        }
        if (cnt != 1)
        {
            cout << "First";
            return 0;
        }
    }
    cout << "Second";
}