Problem:
Score : 600 points

Problem Statement

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.

Sample Input 1

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

Sample Output 1

1
2
3
3
4
4
5
2
2
3

For example, the sequence A obtained from the shortest path from Vertex 1 to Vertex 5 is 1,2,5,3,4. Its longest increasing subsequence is A_1, A_2, A_4, A_5, with the length of 4.

Solution in PyPy3 (7.3.0) (judge verdict: Accepted)

import bisect
import copy
import heapq
import math
import sys
from collections import *
from itertools import accumulate, combinations, permutations, product
from math import gcd
def input():
    return sys.stdin.readline()[:-1]
def ruiseki(lst):
    return [0]+list(accumulate(lst))
sys.setrecursionlimit(1000000)
mod=pow(10,9)+7
al=[chr(ord('a') + i) for i in range(26)]
direction=[[1,0],[0,1],[-1,0],[0,-1]]

n=int(input())
a=list(map(int,input().split()))
uv=[list(map(int,input().split())) for i in range(n-1)]
edge=defaultdict(list)
for i in range(n-1):
    u,v=uv[i]
    edge[u].append(v)
    edge[v].append(u)
# print(edge)

itta=[0]*(n+1)
# lislist=[0]*(n+1)

def dfs(now,lis):
    stk=0
    if a[now-1]>lis[-1]:
        lis.append(a[now-1])
        stk="pop"
    else:
        tmp=bisect.bisect_left(lis, a[now-1])
        stk=[tmp,lis[tmp]]
        lis[tmp]=a[now-1]
    
    # print(now,lis)
    itta[now]=len(lis)
    for i in edge[now]:
        if itta[i]==0:
            dfs(i,lis)
    
    if stk=="pop":
        lis.pop()
    else:
        lis[stk[0]]=stk[1]

dfs(1,[float('inf')])

# print(itta)
for i in range(n):
    print(itta[i+1])