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(int i=0;i<(n);++i)
#define Rep(i,n) for(int i=1;i<=(n);++i)
using LL = long long;
using ULL = unsigned long long;

const int N = 200123;

int ans[N], a[N], s[N], top;
vector<int> vv[N];

void dfs(int u) {
	int tmp = top;
	while(a[u] <= s[tmp]) --tmp;
	int y = s[++tmp];
	s[tmp] = a[u];
	if(top < tmp) top = tmp;
	ans[u] = top;
	for(int v: vv[u]) if(ans[v] == 0) dfs(v);
	s[tmp] = y;
	if(y == 0 && tmp == top) --top;
}

int main()
{
	int n;
	int u, v;
	cin >> n;
	Rep(i, n) scanf("%d", a+i);
	rep(i, n-1) {
		scanf("%d%d", &u, &v);
		vv[u].push_back(v);
		vv[v].push_back(u);
	}
	dfs(1);
	Rep(i, n) cout << ans[i] << endl;
	return 0;
}
