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
import sys
from bisect import bisect_left as bisect
input = sys.stdin.readline
sys.setrecursionlimit(2 * 10**6)


def inpl():
    return list(map(int, input().split()))


class solve:
    def __init__(self, N):
        self.ans = [0] * N
        self.DP = []

    def recur(self, i, pi=-1):
        # DPテーブルを更新し、答えを求める。
        # また、巻き戻す値を覚える。
        rev_i = bisect(self.DP, self.A[i])
        if rev_i == len(self.DP):
            self.DP.append(self.A[i])
            rev_v = None
        else:
            rev_v = self.DP[rev_i]
            self.DP[rev_i] = self.A[i]

        self.ans[i] = len(self.DP)

        for nv in self.edges[i]:
            if nv != pi:
                self.recur(nv, i)

        # 巻き戻す。.
        if rev_v is None:
            self.DP.pop()
        else:
            self.DP[rev_i] = rev_v

        return


def main():
    N = int(input())
    S = solve(N)
    S.A = inpl()
    S.edges = [[] for _ in range(N)]
    for _ in range(N - 1):
        u, v = inpl()
        S.edges[u - 1].append(v - 1)
        S.edges[v - 1].append(u - 1)

    S.recur(0)

    print(*S.ans, sep='\n')
    # print(edges)
    # print(parent)
    # print(DPs)


if __name__ == '__main__':
    main()
