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
#!/usr/bin/env python3
import sys, math, itertools, collections, bisect
input = lambda: sys.stdin.buffer.readline().rstrip().decode('utf-8')
inf = float('inf') ;mod = 10**9+7
mans = inf ;ans = 0 ;count = 0 ;pro = 1
sys.setrecursionlimit(10**6)
n = int(input())
A = list(map(int,input().split()))
G = [[] for i in range(n)]
for i in range(n-1):
  u,v = map(int,input().split())
  u-=1;v-=1
  G[u].append(v)
  G[v].append(u)
dp = [inf] * (n+10)
dp[0] = A[0]
length = 1
fixed = [-1] * (n+10)
ans = [0] * n
def dfs(node,pre,length):
  if node != 0:
    if dp[length-1] < A[node]:
      dp[length] = A[node]
      length += 1
      fixed[node] = inf
    else:
      fixed[node] = dp[bisect.bisect_left(dp,A[node])]
      dp[bisect.bisect_left(dp,A[node])] = A[node]
  ans[node] = length
  for v in G[node]:
    if v == pre:continue
    dfs(v,node,length)
  dp[bisect.bisect_left(dp,A[node])] = fixed[node]

dfs(0,-1,1)
print(*ans,sep = "\n")
    

def LIS(seq):
    dp = [inf]*(n+10)
    dp[0] = seq[0]
    length = 1
    for i in range(1, n):
        if dp[length-1] < seq[i]:
            dp[length] = seq[i]
            length += 1
        else:
            dp[bisect.bisect_left(dp, seq[i])] = seq[i]
    return dp,length