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;

vector<vector<int>> t;
int n, a[222222], L[222222], l[222222];
void g(int u, int v) {
    auto it = lower_bound(l, l + n, a[v]);
    swap(*it, a[v]);
    L[v] = lower_bound(l, l + n, 1e9 + 1) - l;
    for(int w : t[v])
        if(u != w)
            g(v, w);
    *it = a[v];
}
int main() {
    cin >> n;
    for(int i = 0; i < n; i++)
        cin >> a[i + 1];
    t.resize(n + 1);
    for(int i = 1; i < n; i++) {
        int u, v;
        cin >> u >> v;
        t[u].push_back(v);
        t[v].push_back(u);
    }
    fill(l, l + n + 1, 2e9);
    g(0, 1);
    for(int i = 0; i < n; i++)
        cout << L[i + 1] << '\n';
    return 0;
}
