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
#入力高速化 + 再帰回数制限解除
import sys
input=sys.stdin.readline
sys.setrecursionlimit(10000000)

from bisect import bisect_left,bisect_right
n=int(input())
a=list(map(int,input().split()))
edge=[[]for _ in range(n)]
for _ in range(n-1):
  u,v=map(int,input().split())
  u-=1
  v-=1
  edge[u].append(v)
  edge[v].append(u)
ans=[-1]*n
dp=[10**10]*n

#オイラーツアー
#n=頂点数、s=始点、edge=隣接リスト
#木構造を投げると訪問順listが帰ってくる、|ans|=2*n-1
#適宜書き換えて実装を楽にしましょう
def EulerTour(n,s,e):
  EulerTour_list=[]
  def EulerTour_dfs(i,root):
    EulerTour_list.append(i)
    ind=bisect_left(dp,a[i])
    bef=dp[ind]
    dp[ind]=a[i]
    ans[i]=bisect_left(dp,10**10)
    for j in e[i]:
      if j!=root:
        EulerTour_dfs(j,i)
    if root!=-1:
      EulerTour_list.append(root)
      dp[ind]=bef
  EulerTour_dfs(s,-1)
  return EulerTour_list

EulerTour(n,0,edge)
print(*ans)