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
from bisect import bisect_left
from collections import deque
import sys

sys.setrecursionlimit(200001)

N = int(input())

a = list(map(int,input().split()))

graph = [[] for _ in range(N)]

for _ in range(N-1):
    u,v = map(int,input().split())
    u -= 1
    v -= 1
    graph[u].append(v)
    graph[v].append(u)

INF = 10**11
ans = [0]*N
dp = []###INFは空欄として扱って良い
stack = deque([])

def LISonTree(num,pre):
    ###num: 頂点番号
    ###pre: 1つ前にいた頂点番号
    p = bisect_left(dp,a[num])

    ###
    #print(p)
    ###

    if p >= len(dp):
        stack.appendleft((len(dp),INF))
        dp.append(a[num])
    else:
        stack.appendleft((p,dp[p]))
        dp[p] = a[num]
        
    
    q = bisect_left(dp,INF)
    ans[num] = q

    ###
    #print(num,'dp',dp)
    #print(num,'ans',ans)
    ###

    for x in graph[num]:
        if x == pre:
            continue
        LISonTree(x,num)
    
    changed_p, changed_v = stack.popleft()

    dp[changed_p] = changed_v

    ###
    #print(num,'dp',dp)
    ###

LISonTree(0,-1)
#print(ans)

for y in ans:
    print(y)