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 <iostream>
#include <vector>
#include <cmath>
#include <map>
#include <set>
#include <string>
#include <queue>
#include <algorithm>
#include <list>

#define mp make_pair
#define pb push_back

using namespace std;

typedef pair<const int, int> pii;

const int MAXN = 1e5 + 1;
const int inf = 1e9 + 5;
const long double eps = 1e-8;

vector<int> g[MAXN];

void removeNode(map<int, int> &q, int v) {
    for (int i = 0; i < g[v].size(); i++) {
        if (q.find(g[v][i]) != q.end()) {
            q[g[v][i]]--;
        }
    }
    q.erase(v);
}

int main() {
    /*std::ios::sync_with_stdio(false);
    cin.tie(0);*/

    int n;
    cin >> n;
    for (int i = 0; i < n - 1; i++) {
        int a, b;
        cin >> a >> b;
        a--, b--;
        g[a].pb(b);
        g[b].pb(a);
    }

    map<int, int> q;

    for (int i = 0; i < n; i++)
        q[i] = g[i].size();

    while (!q.empty()) {
        pii p = *(q.begin());
        if (p.second == 0) {
            cout << "First";
            return 0;
        }
        if (p.second > 1) {
            cout << "Second";
            return 0;
        }
        for (int i = 0; i < g[p.first].size(); i++) {
            if (q.find(g[p.first][i]) != q.end()) {
                removeNode(q, g[p.first][i]);
            }
        }
        q.erase(p.first);
    }

    cout << "Second";

    return 0;
}