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
input = sys.stdin.readline
sys.setrecursionlimit(10 ** 6)
from bisect import bisect_left
 
N = int(input())
a = [int(i) for i in input().split()]
vec = [[] for _ in range(N)]
for _ in range(N - 1) :
  u, v = map(int, input().split())
  vec[u - 1].append(v - 1)
  vec[v - 1].append(u - 1)
 
lis = [a[0]]
ret = [0] * N
def dfs(cur, pre) :
  for nex in vec[cur] :
    if nex == pre : continue
    if a[nex] > lis[-1] :
      lis.append(a[nex])
      ret[nex] = len(lis)
      dfs(nex, cur)
      lis.pop()
    else :
      pos = bisect_left(lis, a[nex])
      v = lis[pos]
      lis[pos] = a[nex]
      ret[nex] = len(lis)
      dfs(nex, cur)
      lis[pos] = v
 
dfs(0, -1)
 
ret[0] = 1
for i in range(N) :
  print(ret[i])