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
def main():
    import sys
    input = sys.stdin.readline

    from bisect import bisect_left

    N = int(input())
    A = list(map(int, input().split()))

    G = [[] for _ in range(N)]

    for _ in range(N - 1):
        u, v = map(int, input().split())
        u -= 1
        v -= 1
        G[u].append(v)
        G[v].append(u)

    root = 0
    stack = [root]
    visited = set()
    visited.add(root)
    done = set()
    vs = []

    ans = [0] * N
    memo = [None] * N
    memo[root] = (0, A[root])

    INF = 10 ** 10
    dp = [INF] * N

    while stack:
        now_ = stack[-1]
        if now_ in done:
            j, a = memo[now_]
            dp[j] = a
            stack.pop()
            vs.append(now_)
        else:
            for next_ in G[now_][::-1]:
                if next_ in visited:
                    continue
                visited.add(next_)
                stack.append(next_)
            done.add(now_)

            tmp = bisect_left(dp, A[now_])
            memo[now_] = (tmp, dp[tmp])
            dp[tmp] = A[now_]
            ans[now_] = bisect_left(dp, INF)

    # print (vs)
    print (*ans, sep = '\n')

if __name__ == '__main__':
    main()