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
read = sys.stdin.buffer.read
readline = sys.stdin.buffer.readline
readlines = sys.stdin.buffer.readlines
from collections import deque
from bisect import bisect_left

n = int(readline())
A = list(map(int,readline().split()))
g = [[] * n for i in range(n)]
for i in range(n-1):
    u, v = map(int, readline().split())
    u -= 1
    v -= 1
    g[u].append(v)
    g[v].append(u)
    
que = deque()
inf = 10**9+1
dp = [inf]*n
que.append((0,-1))
chk = [False]*n
chk[0] = True
cnt = 0
ans = [1]*n

while que:
  now,d = que.pop()
  if d != -1:
    dp[now] = d
  else:
    ind = bisect_left(dp,A[now])
    que.append((ind,dp[ind]))
    dp[ind] = A[now]
    ans[now] = bisect_left(dp,inf)
    cnt += 1
    for i in g[now]:
      if not chk[i]:
        chk[i] = True
        que.append((i,-1))
  if cnt == n:
    break
        
print(*ans,sep='\n')