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
from bisect import bisect_left


def main():
    N = int(input())
    A = list(map(int, input().split()))
    adj = [set() for _ in range(N)]
    for i in range(N-1):
        u, v = map(int, input().split())
        adj[u-1].add(v-1)
        adj[v-1].add(u-1)
    ans = [0] * N
    q = [(0, -1, [], 0)]
    history = []
    while len(q):
        cur, prev, lis, depth = q.pop()
        while depth < len(history):
            pos, val = history.pop()
            if val is None:
                lis.pop()
            else:
                lis[pos] = val
        idx = bisect_left(lis, A[cur])
        if idx == len(lis):
            history.append((idx, None))
            lis.append(A[cur])
        else:
            history.append((idx, lis[idx]))
            lis[idx] = A[cur]
        ans[cur] = len(lis)
        for nxt in adj[cur]:
            if nxt == prev:
                continue
            q.append((nxt, cur, lis, depth+1))
    for i in range(N):
        print(ans[i])


if __name__ == "__main__":
    main()
