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
import bisect

N=int(input())
A=list(map(int,input().split()))
ans=[0]*N
Edge=[[] for _ in range(N)]
loute=[str(0)]*N
for i in range(N-1):
    u,v=map(int,input().split())
    Edge[u-1].append(v-1)
    Edge[v-1].append(u-1)
  
tmp=deque(str(0))
CHN=deque()
LIS=[10**18]*N
label=[False]*N
while tmp:
    T0=int(tmp[-1])
    if label[T0]:
        tmp.pop()
        pos,Val=CHN.pop()
        LIS[pos]=Val
        continue

    pos=bisect.bisect_left(LIS,A[T0])
    CHN.append((pos,LIS[pos]))
    LIS[pos]=A[T0]
    ans[T0]=bisect.bisect_left(LIS,10**18)

    if Edge[T0]:
        for i in Edge[T0]:
            if label[i]==False:
                tmp.append(i)
        label[T0]=True
    else:
        label[T0]=True
    
for i in range(N):
    print(ans[i])