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;
const int N = 2e5 + 5, inf = 1e9 + 7;
int a[N], dp[N], ans[N], n;
vector <int> g[N];

void dfs(int v, int p = 0) {
    int x = lower_bound(dp, dp + n + 1, a[v]) - dp, val = dp[x];
    dp[x] = a[v], ans[v] = lower_bound(dp, dp + n + 1, inf) - dp;
    for (int u : g[v]) if (u != p) dfs(u, v);
    dp[x] = val;
}

int main() {
    cin >> n;
    for (int i = 1; i <= n; i++) cin >> a[i];
    for (int i = 0; i < n - 1; i++) {
        int u, v; cin >> u >> v;
        g[u].push_back(v), g[v].push_back(u);
    }
    fill(dp, dp + n + 1, inf), dfs(1);
    for (int i = 1; i <= n; i++) cout << ans[i] << "\n";

    return 0;
}
