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
n=int(input())
A=list(map(int,input().split()))
g=[[]for i in range(n)]
for i in range(n-1):
    a,b=map(int,input().split())
    a-=1
    b-=1
    g[a].append(b)
    g[b].append(a)

import sys
sys.setrecursionlimit(10**6)

lastdp=[10**9+1]*n
recode=[[10**9+1]for i in range(n)]
ans=[0]*n
from bisect import bisect_left
def rec(me,parent):
    global lastdp
    a=A[me]
    point=bisect_left(lastdp,a)
    recode[point].append(a)
    lastdp[point]=a
    tmpans=bisect_left(lastdp,10**9+1)
    ans[me]=tmpans

    for child in g[me]:
        if child!=parent:
            rec(child,me)

    recode[point].pop()
    lastdp[point]=recode[point][-1]
    return

rec(0,-1)
print(*ans,sep='\n')
