We have a tree with N vertices, whose i-th edge connects Vertex u_i and Vertex v_i. Vertex i has an integer a_i written on it. For every integer k from 1 through N, solve the following problem:

* We will make a sequence by lining up the integers written on the vertices along the shortest path from Vertex 1 to Vertex k, in the order they appear. Find the length of the longest increasing subsequence of this sequence.



Here, the longest increasing subsequence of a sequence A of length L is the subsequence A_{i_1} , A_{i_2} , ... , A_{i_M} with the greatest possible value of M such that 1 \leq i_1 < i_2 < ... < i_M \leq L and A_{i_1} < A_{i_2} < ... < A_{i_M}.

Constraints

* 2 \leq N \leq 2 \times 10^5
* 1 \leq a_i \leq 10^9
* 1 \leq u_i , v_i \leq N
* u_i \neq v_i
* The given graph is a tree.
* All values in input are integers.

Input

Input is given from Standard Input in the following format:


N
a_1 a_2 ... a_N
u_1 v_1
u_2 v_2
:
u_{N-1} v_{N-1}


Output

Print N lines. The k-th line, print the length of the longest increasing subsequence of the sequence obtained from the shortest path from Vertex 1 to Vertex k.

Example

Input

10
1 2 5 3 4 6 7 3 2 4
1 2
2 3
3 4
4 5
3 6
6 7
1 8
8 9
9 10


Output

1
2
3
3
4
4
5
2
2
3
#include<bits/stdc++.h>
using namespace std;
int dp[200005];
vector<int> G[200005];
int n;
int a[200005];
int ans[200005];
const int inf = 0x3f3f3f3f;
void dfs(int u,int p){
	int ind = (int)(lower_bound(dp,dp+n,a[u]) - dp);
	int cnt = dp[ind];
	dp[ind] = a[u];
	ans[u] = (int)(lower_bound(dp,dp+n,inf) - dp);
	for(int i:G[u]){
		if(i!=p){
			dfs(i,u);
		}
	}
	//undo
	dp[ind] = cnt;
}
int main(){
	cin>>n;
	for(int i=0;i<n;i++){
		cin>>a[i];
	}
	for(int i=0;i<n-1;i++){
		int t1,t2;
		cin>>t1>>t2;
		t1--,t2--;
		G[t1].push_back(t2);
		G[t2].push_back(t1);
	}
	fill(dp,dp+n+2,inf);
	dfs(0,-1);
	for(int i=0;i<n;i++) cout<<ans[i]<<endl;
	return 0;
}