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 <iostream>
#include <vector>

using namespace std;

const int MAX = 2e5 + 1;

int N, a[MAX], d[MAX], ans[MAX];
vector<int> v[MAX];

void calc(int p, int q){
	if(ans[p]){
		return;
	}
	int k = lower_bound(d, d + q, a[p]) - d;
	if(k == q){
		q++;
	}
	ans[p] = q;

	int temp = d[k];
	d[k] = a[p];
	for(auto t: v[p]){
		calc(t, q);
	}
	d[k] = temp;
}

int main(){
	cin >> N;
	for (int i = 1; i <= N; i++){
		cin >> a[i];
	}
	for (int i = 0; i < N - 1; i++){
		int tu, tv;
		cin >> tu >> tv;
		v[tu].push_back(tv);
		v[tv].push_back(tu);
	}
	for (int i = 0; i < N; i++){
		d[i] = 1e9;
	}

	calc(1, 0);
	for (int i = 1; i <= N; i++){
		cout << ans[i] << endl;
	}
	return 0;
}
