Problem: 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 edge is painted blue. Takahashi will convert this blue tree into a red tree, by performing the following operation N-1 times:

* Select a simple path that consists of only blue edges, and remove one of those edges.
* Then, span a new red edge between the two endpoints of the selected path.



His objective is to obtain a tree that has a red edge connecting vertices c_i and d_i, for each i.

Determine whether this is achievable.

Constraints

* 2 ≤ N ≤ 10^5
* 1 ≤ a_i,b_i,c_i,d_i ≤ N
* a_i ≠ b_i
* c_i ≠ d_i
* Both input graphs are trees.

Input

Input is given from Standard Input in the following format:


N
a_1 b_1
:
a_{N-1} b_{N-1}
c_1 d_1
:
c_{N-1} d_{N-1}


Output

Print `YES` if the objective is achievable; print `NO` otherwise.

Examples

Input

3
1 2
2 3
1 3
3 2


Output

YES


Input

5
1 2
2 3
3 4
4 5
3 4
2 4
1 4
1 5


Output

YES


Input

6
1 2
3 5
4 6
1 6
5 1
5 3
1 4
2 6
4 3
5 6


Output

NO

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;

const int MaxN=1e5+5;

int N,H=1,T=1;
int Fa[MaxN];

pair<int,int> Q[MaxN<<1];
set<int> To[MaxN];
map<int,int> Map[MaxN];
set<int>::iterator it1,it2;

int Get_Fa(int x){
	return x==Fa[x]?x:Fa[x]=Get_Fa(Fa[x]);
}

void Add(int u,int v){
	if(u>v)
		swap(u,v);
	To[u].insert(v);
	To[v].insert(u);
	if(++Map[u][v]>1)
		Q[T++]=make_pair(u,v);
}

void Del(int u,int v){
	if(u>v)
		swap(u,v);
	To[u].erase(v);
	To[v].erase(u);
	Map[u][v]=0;
}

int main(){
	int i,u,v,r1,r2;
	scanf("%d",&N);
	for(i=1;i<=N-1<<1;i++){
		scanf("%d%d",&u,&v);
		Add(u,v);
	}
	for(i=1;i<=N;i++)
		Fa[i]=i;
	for(i=1;i<N;i++){
		if(H==T){
			puts("NO");
			return 0;
		}
		u=Q[H].first,v=Q[H].second,H++;
		if(To[r1=Get_Fa(u)].size()>To[r2=Get_Fa(v)].size())
			swap(r1,r2);
		Fa[r1]=r2;
		Del(r1,r2);
		for(it1=To[r1].begin();it1!=To[r1].end();it1=it2){
			it2=it1;
			it2++;
			Del(r1,*it1);
			Add(r2,*it1);
		}
	}
	puts("YES");
	return 0;
}