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
from bisect import bisect_left

sys.setrecursionlimit(10 ** 7)
rl = sys.stdin.readline


def solve():
    INF = 10 ** 18
    N = int(rl())
    A = list(map(int, rl().split()))
    graph = [[] for _ in range(N)]
    for _ in range(N - 1):
        u, v = map(lambda x: int(x) - 1, rl().split())
        graph[u].append(v)
        graph[v].append(u)
    
    ans = [-1] * N
    stack = [0]
    dp = [INF] * (N + 1)
    rewind = [(-1, -1) for _ in range(N)]
    while stack:
        node = stack.pop()
        if 0 <= node:
            idx = bisect_left(dp, A[node])
            rewind[node] = (idx, dp[idx])
            dp[idx] = A[node]
            ans[node] = bisect_left(dp, INF)
            stack.append(~node)
            for child in graph[node]:
                if ans[child] == -1:
                    stack.append(child)
        else:
            idx, val = rewind[~node]
            dp[idx] = val
    print(*ans, sep='\n')


if __name__ == '__main__':
    solve()
