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;
typedef long long ll;
typedef pair<int,int> P;
#define rep(i,m,n) for(int i=m;i<n;i++)
ll mod=1e9+7;

int n;
int a[200000];
vector<int> tree[200000];
int ans[200000];
int d[200000];

void dfs(int i){
  int k=lower_bound(d,d+n,a[i])-d;
  int old=d[k];
  d[k]=a[i];
  ans[i]=lower_bound(d,d+n,2e9)-d;
  
  rep(j,0,tree[i].size()){
    if(ans[tree[i][j]]==0){
      dfs(tree[i][j]);
    }
  }
  d[k]=old;
}

int main(){
  cin>>n;
  rep(i,0,n) cin>>a[i];
  rep(i,0,n-1){
    int u,v;
    cin>>u>>v;
    tree[u-1].push_back(v-1);
    tree[v-1].push_back(u-1);
  }
  rep(i,0,n) d[i]=2e9;
  rep(i,0,n) ans[i]=0;
  dfs(0);
  rep(i,0,n) cout<<ans[i]<<endl;
}