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
import time

readline = sys.stdin.readline
MOD = 10 ** 9 + 7
INF = float('INF')
sys.setrecursionlimit(10**5)


def main():
    n = int(input())
    a = list(map(int, readline().split()))
    node = {i: [] for i in range(n)}

    for _ in range(n - 1):
        u, v = map(int, readline().split())
        u, v = u - 1, v - 1
        node[u].append(v)
        node[v].append(u)

    ans = [0] * n
    dp = [INF] * n
    dp[0] = a[0]
    visited = [False] * n

    from collections import deque
    import bisect

    def dfs(cur):
        visited[cur] = True
        nv = a[cur]
        index = bisect.bisect_left(dp, nv)
        pv = dp[index]
        dp[index] = nv
        suc = node[cur]

        for nx in suc:
            if not visited[nx]:
                dfs(nx)

        ans[cur] = bisect.bisect_left(dp, INF)
        dp[index] = pv

    dfs(0)
    print(*ans, sep="\n")


if __name__ == '__main__':
    main()
