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: Time Limit Exceeded)

#include <bits/stdc++.h>
using namespace std;
const int N = 1e5 + 5;
int n, cnt;
vector <int> a[N];
bool vis[N] = {0};


int cou(int u) {
    int res = 0;
    for (int i = 0; i < a[u].size(); i++) {
        int v = a[u][i];
        if (a[v].size() == 1) res++;
    }
    return res;
}

int dfs(int u) {
    vis[u] = true;
    cnt++;
    for (int i = 0; i < a[u].size(); i++) {
        int v = a[u][i];
        if (!vis[v]) dfs(v);
    }
}

int main() {
    scanf("%d", &n);
    for (int i = 1; i < n; i++) {
        int x, y; scanf("%d %d", &x, &y);
        a[x].push_back(y);
        a[y].push_back(x);
    }
    for (int i = 1; i <= n; i++) {
        if (cou(i) >= 2) {
            cout << "First" << endl;
            return 0;
        }
    }
    if (n & 1) {
        cout << "First" << endl;
        return 0;
    }
    vector <int> b;
    for (int i = 1; i <= n; i++) {
        if (a[i].size() >= 4) b.push_back(i);
    }
    int dem = 0;
    for (int t = 0; t < b.size(); t++) {
        int i = b[t];
        dem += n;
        memset(vis, false, sizeof(vis));
        vis[i] = true;
        int res = 0;
        for (int j = 0; j < a[i].size(); j++) {
            cnt = 0;
            dfs(a[i][j]);
            res += cnt % 2;
        }
        if (res >= 2) {
            cout << "First" << endl;
            return 0;
        }
        if (dem >= 125000000) break;
    }
    srand(time(NULL));
    cout << "Second" << endl;
}