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
    sys.setrecursionlimit(10**9)
    input = sys.stdin.readline
    from collections import deque
    from bisect import bisect_left

    N = int(input())
    a = list(map(int, input().split()))
    tree = [[] for _ in [0]*N]
    for u, v in [map(int, input().split()) for _ in [0]*(N-1)]:
        tree[u-1].append(v-1)
        tree[v-1].append(u-1)
    
    dp = [1001001001] * (N+1)
    dp[0] = -1001001001
    ans = [0] * N
    def dfs(now, p=-1):
        idx = bisect_left(dp, a[now])
        old = dp[idx]
        dp[idx] = a[now]
        ans[now] = idx
        if p != -1 and ans[now] < ans[p]:
            ans[now] = ans[p]
        for child in tree[now]:
            if child == p: continue
            dfs(child, now)
        dp[idx] = old
    dfs(0)

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

main()