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(10**9)
n = int(input())
a = [0]+list(map(int,input().split()))
tree = [[] for _ in range(n+1)]
for _ in range(n-1):
  u,v = map(int,input().split())
  tree[u].append(v)
  tree[v].append(u)
#print(tree)
d = [float("inf")]*(n+1)
ans = [0]*(n+1)

def dfs(now, pre):
  i = bisect.bisect_left(d, a[now])
  tmp = d[i]
  d[i] = a[now]
  ans[now] = bisect.bisect_left(d, float("inf"))
  for nxt in tree[now]:
    if nxt == pre:
      continue
    dfs(nxt, now)
  d[i] = tmp
dfs(1,-1)

for i in ans[1:]:
  print(i)