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;
#define rep(i,n) for(ll i=0;i<n;i++)
#define all(v) v.begin(), v.end()
typedef long long ll;
typedef pair<ll,ll> P;
typedef vector<ll> vec;
typedef vector<vec> mat;
int r[200001]={0};
ll inf=1e18;
vec G[200001],A(200001),dp(200001,inf),ans(200001);
int dfs(int x){
    if(r[x]) return 0;
    r[x]=1;
    ll a=lower_bound(all(dp),A[x])-dp.begin(),b=*lower_bound(all(dp),A[x]);
    *lower_bound(all(dp),A[x])=A[x];
    ans[x]=lower_bound(all(dp),inf)-dp.begin();
    for(auto i:G[x]) dfs(i);
    dp[a]=b;
    return 0;
}
int main(){
    int n,u,v;
    cin>>n;
    rep(i,n) cin>>A[i+1];
    rep(i,n-1) cin>>u>>v,G[u].push_back(v),G[v].push_back(u);
    dfs(1);
    rep(i,n) cout<<ans[i+1]<<"\n";
}