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 MaxN = 2e5+10;
int N;
int a[MaxN];
vector<int> adj[MaxN];
int dp[MaxN];
int len[MaxN];


void dfs(int u, int fa, int maxlen)
{
    int k = lower_bound(dp, dp + N, a[u]) - dp;
    len[u] = max(maxlen, k + 1);
    int dpk_old = dp[k];
    for (int v : adj[u]) {
        if (v != fa) {
            dp[k] = a[u];
            dfs(v, u, max(maxlen, k + 1));
            dp[k] = dpk_old;
        }
    }
}


void solve()
{
    memset(dp, 0x7f, sizeof(dp));
    dfs(1, 0, 0);
    for (int i = 1; i <= N; i++) cout << len[i] << endl;
}


int main()
{
    ios::sync_with_stdio(false); cin.tie(0);

    cin >> N;
    for (int i = 1; i <= N; i++) cin >> a[i];
    int u, v;
    for (int i = 1; i <= N - 1; i++) {
        cin >> u >> v;
        adj[u].push_back(v);
        adj[v].push_back(u);
    }
    solve();
    return 0;
}
