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
readline = sys.stdin.readline

sys.setrecursionlimit(10 ** 7)


def main():
    N = int(input())
    As = list(map(int, input().split()))

    edges = (map(int, readline().strip().split()) for _ in range(N-1))

    tree = [[] for _ in range(N)]
    for u, v in edges:
        tree[u-1].append(v-1)
        tree[v-1].append(u-1)

    inf = 10 ** 18
    dp = [inf] * (N+1)
    ans = [0] * N

    def dfs(node, parent):
        v = As[node]
        lb = binary_search(dp, v)
        old = dp[lb]
        dp[lb] = v
        ans[node] = binary_search(dp, inf)
        for child in tree[node]:
            if child == parent:
                continue
            dfs(child, node)
        dp[lb] = old

    dfs(0, 0)

    for k in range(N):
        print(ans[k])


def binary_search(seq, v):
    left = 0
    right = len(seq) - 1
    center = right // 2

    while left != right:
        if v <= seq[center]:
            right = center
        else:
            left = center+1
        center = (right + left) // 2

    return center


if __name__ == "__main__":
    main()
