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: Wrong Answer)

import sys
sys.setrecursionlimit(1000000)
def input():
    return sys.stdin.readline()
n = int(input())
a = [int(i) for i in input().split()]
e = [[] for _ in range(n)]
for _ in range(n - 1):
    aa, bb = [int(i) - 1 for i in input().split()]
    e[aa].append(bb)
    e[bb].append(aa)

ans = [0] * n
ans[0] = 1
ans2 = [0] * n
ans2[0] = 1


def dfs(i=0, r=-1, x=[], y=[]):
    if r == -1:
        x.append(a[i])
    else:
        if x[0] >= a[i]:
            ans[i] = 1
        else:
            mi = 0
            ma = len(x) - 1
            while mi != ma:
                mm = (mi + ma) // 2+1
                if x[mm] < a[i]:
                    mi = mm
                else:
                    ma = mm - 1
            ans[i] = ans[y[mi]]+1
        ans2[i] = max(ans2[r], ans[i])
        x.append(max(x[-1], a[i]))
    y.append(i)
    for j in e[i]:
        if j != r:
            dfs(j, i)
    y.pop()
    x.pop()


dfs()
for i in ans2:
    print(i)