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
import sys
sys.setrecursionlimit(1<<30)
def dfs(i,LIS):
    if a[i] > LIS[-1]:
        LIS.append(a[i])
        ans[i] = len(LIS)
        for j in edges[i]:
            if parent[i] != j:
                parent[j] = i
                dfs(j,LIS)
        LIS.pop()
    else:
        index = bisect.bisect_left(LIS, a[i])
        x = LIS[index]
        LIS[index] = a[i]
        ans[i] = len(LIS)
        for j in edges[i]:
            if parent[i] != j:
                parent[j] = i
                dfs(j,LIS)
        LIS[index] = x

N = int(input())
a = [0]+list(map(int,input().split()))
edges = [[] for _  in range(N+1)]
ans = [-1]*(N+1)
parent = [-1]*(N+1)
for i in range(N-1):
    u,v = map(int,input().split())
    edges[u].append(v)
    edges[v].append(u)
LIS = [a[1]]
dfs(1,LIS)
for i in ans[1:]:
    print(i)