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>
#define rep(i, n) for(int i = 0; i < int(n); i++)
#define all(in) in.begin(),in.end()
using namespace std;

const int INF = 1<<30;
vector<int> num, dp, ans;
vector<vector<int>> G;

void dfs(int u){
    int i = lower_bound(all(dp), num[u]) - dp.begin();
    int save = dp[i]; dp[i] = num[u];
    ans[u] = lower_bound(all(dp), INF) - dp.begin();
    for(auto next : G[u]) if(ans[next] == 0) dfs(next);
    dp[i] = save;
}

int main(){
    int n; cin >> n;
    num.assign(n, 0); rep(i,n) cin >> num[i];
    dp.assign(n, INF);
    ans.assign(n, 0);
    G.assign(n, vector<int>());
    rep(i,n-1){
        int a, b; cin >> a >> b;
        a--; b--;
        G[a].push_back(b);
        G[b].push_back(a);
    }

    dfs(0);

    rep(i,n) cout << ans[i] << endl;
}