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>

typedef long long ll;
typedef unsigned long long ull;

using namespace std;

const int MAXN=1E5+10;

int n,fa[MAXN];
set<int> S[MAXN];
map<ll,int> M;
queue<pair<int,int> > q;

int find(int x){return x==fa[x]?x:fa[x]=find(fa[x]);}

ll get(int u,int v)
{
	if(u>v)
		swap(u,v);
	return 1ll*MAXN*u+v;
}

void Add_Edge(int u,int v)
{
	S[u].insert(v);
	S[v].insert(u);
	if(++M[get(u,v)]==2)
		q.emplace(u,v);
}

void Erase_Edge(int u,int v)
{
	S[u].erase(v);
	S[v].erase(u);
	M.erase(get(u,v));
}

int main()
{
	scanf("%d",&n);
	for(int i=1;i<=n;i++)
		fa[i]=i;
	for(int i=1,u,v;i<=2*n-2;i++)
	{
		scanf("%d%d",&u,&v);
		Add_Edge(u,v);
	}
	for(int i=1;i<n;i++)
	{
		if(q.empty())
			return puts("NO"),0;
		int x=find(q.front().first),y=find(q.front().second);
		q.pop();
		if(S[x].size()<S[y].size())
			swap(x,y);
		Erase_Edge(x,y);
		for(set<int>::iterator it=S[y].begin(),nxt;it!=S[y].end();it=nxt)
		{
			int v=*it;
			nxt=it;nxt++;
			Erase_Edge(y,v);
			Add_Edge(x,v);
		}
		fa[y]=x;
	}
	puts("YES");
}