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: Wrong Answer)

def getN():
    return int(input())
def getNM():
    return map(int, input().split())
def getList():
    return list(map(int, input().split()))

from collections import defaultdict, deque
from sys import exit
import math
import copy
from bisect import bisect_left

import sys
sys.setrecursionlimit(1000000)

def calc(arr, query):
    ret = 0
    # print(arr)\
    for q in query:
        if arr[q[1] - 1] - arr[q[0] - 1] == q[2]:
            ret += q[3]

    return ret

def dfs(tree, nums, answers, current, LIS, visited):
    # if current >= 7:
        # print(current, LIS)
    visited[current] = 1
    nc = nums[current]
    if nc > LIS[-1]:
        LIS.append(nc)
    else:
        ins = bisect_left(LIS, nc)
        if LIS[ins] != nc:
            if ins == 0:
                LIS[ins] = nc
            elif LIS[ins-1] != nc:
                LIS[ins] = nc
    answers[current] = len(LIS)
    # 分岐するときはLIS複製
    # そうでなければポインタ私
    if len(tree[current]) == 1:
        return
    if len(tree[current]) == 2 and current != 0:
        for nx in tree[current]:
            if not visited[nx]:
                dfs(tree, nums, answers, nx, LIS, visited)
        return
    else:
        org = LIS
        # print("bunki", current)
        for nx in tree[current]:
            if not visited[nx]:
                LIScp = copy.copy(org)
                dfs(tree, nums, answers, nx, LIScp, visited)

        return



def main():
    n = getN()
    nums = getList()
    tree = [[] for _ in range(n)]
    for i in range(n - 1):
    # start, end, cost
        a, b = getList()
        # 0-indexed
        a -= 1
        b -= 1
        tree[a].append(b)
        tree[b].append(a)

    # print(tree)
    answers = [0 for i in range(n)]
    answers[0] = 1
    LIS = [nums[0]]
    visited = [0 for i in range(n)]
    visited[0] = 1
    dfs(tree, nums, answers, 0, LIS, visited)

    for ans in answers:
        print(ans)

if __name__ == "__main__":
    main()