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 collections import deque
from bisect import bisect_left as bile

def main():
    import sys
    def input(): return sys.stdin.readline().rstrip()
    sys.setrecursionlimit(10**6)
    n = int(input())
    arr = list(map(int, input().split()))
    adj = [[] for _ in range(n)]
    for i in range(n-1):
        a, b = map(int, input().split())
        a -= 1
        b -= 1
        adj[a].append(b)
        adj[b].append(a)

    ans = [0]*n
    inf = 1e18
    dp = [inf]*(n+10)
    parent = [-1]*n
    def dfs(s):
        idx = bile(dp, arr[s])
        tmp = dp[idx]
        dp[idx] = arr[s]
        ans[s] = bile(dp, inf)

        for v in adj[s]:
            if v == parent[s]:
                continue
            parent[v] = s
            dfs(v)
        dp[idx] = tmp
        return ans
        
    ans = dfs(0)
    for a in ans:
        print(a)


if __name__ == '__main__':
    main()