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 bisect

def dfs(graph, root, A, res, lis):
    stack = [(root,)]
    while stack:
        # print(stack)
        state = stack[-1]
        del stack[-1]
        u = state[0]
        if res[u] == -1:
            a = A[u]
            p = bisect.bisect_left(lis, a)
            if p == len(lis):
                stack.append((u, -1))
                lis.append(a)
            else:
                stack.append((u, p, lis[p]))
                lis[p] = a
            res[u] = len(lis)
            for v in graph[u]:
                if res[v] == -1:
                    stack.append((v,))
        else:
            p = state[1]
            if p >= 0:
                lis[p] = state[2]
            else:
                del lis[-1]

n = int(input())
A = [int(x) for x in input().split()]
graph = [[] for i in range(n)]
for i in range(n - 1):
    u, v = [int(x) - 1 for x in input().split()]
    graph[u].append(v)
    graph[v].append(u)
res = [-1] * n
dfs(graph, 0, A, res, [])
print(*res, sep='\n')