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
# -*- coding: utf-8 -*-
import sys
from bisect import bisect_left,bisect_right
sys.setrecursionlimit(10**9)
INF=10**18
MOD=10**9+7
input=lambda: sys.stdin.readline().rstrip()
YesNo=lambda b: bool([print('Yes')] if b else print('No'))
YESNO=lambda b: bool([print('YES')] if b else print('NO'))
int1=lambda x:int(x)-1

def main():
    N=int(input())
    a=list(map(int,input().split()))
    edge=[[] for _ in range(N)]
    for _ in range(N-1):
        x,y=map(int1,input().split())
        edge[x].append(y)
        edge[y].append(x)
    stack=[(0,1)]
    used=[False]*N
    ans=[0]*N
    dp=[INF]*(N+1)
    dp[0]=0
    dpdp=[]
    prev_l=1
    def dfs(v,l):
        used[v]=True
        index=bisect_right(dp,a[v]-1,hi=l)
        dpdp=(index,dp[index])
        dp[index]=a[v]
        ans[v]=bisect_left(dp,INF,hi=l+1)-1
        for nv in edge[v]:
            if not used[nv]:
                i,x=dfs(nv,l+1)
                dp[i]=x
        return dpdp
    dfs(0,1)
    for a in ans:
        print(a)

if __name__ == '__main__':
    main()
