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
import bisect


def dfs(edge):
    insert = bisect.bisect_left(lis_array, a_array[edge - 1])
    old = lis_array[insert]
    lis_array[insert] = a_array[edge - 1]
    ans = bisect.bisect_left(lis_array, INF)
    ans_array[edge - 1] = ans
    for next_edge in node_dic[edge]:
        if use_array[next_edge - 1] == -1:
            use_array[next_edge - 1] = 1
            dfs(next_edge)
    lis_array[insert] = old


sys.setrecursionlimit(10 ** 9)
N = int(input())
a_array = list(map(int, input().split()))

node_dic = {}
for _ in range(N - 1):
    u, v = map(int, input().split())
    if u in node_dic:
        node_dic[u].append(v)
    else:
        node_dic[u] = [v]

    if v in node_dic:
        node_dic[v].append(u)
    else:
        node_dic[v] = [u]

# print(node_dic)

use_array = [-1] * N
ans_array = [0] * N

INF = 2 * 10 ** 9
lis_array = [INF] * N

use_array[0] = 1
dfs(1)
print("\n".join(map(str, ans_array)))
