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
import sys
def main():
    input = sys.stdin.readline
    N = int(input())
    *A, = map(int, input().split())
    G = [[] for _ in range(N)]
    for _ in range(N - 1):
        u, v = map(int, input().split())
        u, v = u - 1, v - 1
        G[u].append(v)
        G[v].append(u)

    INF = 10**10
    tbl = [INF] * N
    stk = [(0, 0)]
    ans = [0] * N
    mx = -1
    while stk:
        s, a = stk.pop()
        if s == 0:
            v = a
            i = bisect_left(tbl, A[v])
            mx = max(mx, i)
            ans[v] = mx + 1
            stk.append((1, (i, tbl[i])))
            tbl[i] = A[v]
            for to in G[v]:
                if ans[to] > 0: continue
                stk.append((0, to))
        else:
            idx, val = a
            tbl[idx] = val
            if val == INF: mx = idx - 1
    for a in ans:
        print(a)

if __name__ == '__main__':
    main()
