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 Python (3.8.2) (judge verdict: Runtime Error)

# coding: utf-8
import sys
from bisect import bisect_left, bisect_right, insort
sys.setrecursionlimit(10 ** 7)

sr = lambda: sys.stdin.readline().rstrip()
ir = lambda: int(sr())
lr = lambda: list(map(int, sr().split()))

N = ir()
A = [0] + lr()  # 1-indexed
graph = [[] for _ in range(N+1)]  # 1-indexed
for _ in range(N-1):
    a, b = lr()
    graph[a].append(b)
    graph[b].append(a)

answer = [0] * (N+1)  # 1-indexed
INF = 10 ** 6
parent = [-1] * (N+1)

def dfs(n, arr):
    """arrはLISのリスト、answerのリストを更新してから子にarrを渡す"""
    i = bisect_left(arr, INF)
    answer[n] = i
    for c in graph[n]:
        if c == parent[n]:
            continue
        parent[c] = n
        new = arr[:]
        i = bisect_left(new, A[c])
        new[i] = A[c]
        dfs(c, new)

B = [INF] * (N+1)
B[0] = A[1]
dfs(1, B)
for a in answer[1:]:
    print(a)