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 <stdio.h>
#include <vector>
#include <algorithm>
using namespace std;
#define PB push_back

constexpr int kN = int(2E5 + 10), kInf = int(2E9 + 10);
vector<int> graph[kN];
int ans[kN], p[kN], a[kN], n, nsz;
void dfs(int pos, int from = 1) {
	int l = lower_bound(p, p + n, a[pos]) - p, tmp;

	tmp = p[l];
	p[l] = a[pos];

	if (tmp == kInf) nsz++;
	ans[pos] = nsz;

	for (int i : graph[pos]) if (i != from) dfs(i, pos);	
	
	p[l] = tmp;
	if (tmp == kInf) nsz--;
	return ;
}

int main() {
	int l, r;
	scanf("%d", &n);
	for (int i = 1; i <= n; i++) scanf("%d", &a[i]);
	for (int i = 1; i < n; i++) {
		scanf("%d%d", &l, &r);
		graph[l].PB(r);
		graph[r].PB(l);
	}
	for (int i = 1; i < kN; i++) p[i] = kInf;
	nsz = p[0] = 0;
	dfs(1);

	for (int i = 1; i <= n; i++) printf("%d\n", ans[i]);
}
