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: Runtime Error)

n = int(input())
a = list(map(int, input().split()))
uv = [list(map(int, input().split())) for _ in range(n - 1)]

import sys
sys.setrecursionlimit(10 ** 7)

from collections import defaultdict
g = defaultdict(list)
for i in range(n - 1):
    u = uv[i][0] - 1
    v = uv[i][1] - 1
    g[u].append(v)
    g[v].append(u)

from bisect import bisect_left
ans = [0] * n
visited = [0] * n

def dfs(cur):
    visited[cur] = 1
    idx = bisect_left(dp, a[cur])
    #print(cur, dp, idx)

    if idx == len(dp):
        old = float('inf')
        dp.append(a[cur])
    else:
        old = dp[idx]
        dp[idx] = a[cur]

    ans[cur] = len(dp)

    for target in g[cur]:
        if visited[target] == 0:
            dfs(target)

            if old == float('inf'):
                dp.pop()
            else:
                dp[idx] = old

dp = []
dfs(0)

for _ans in ans:
    print(_ans)