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.

Example

Input

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


Output

1
2
3
3
4
4
5
2
2
3
from sys import stdin
import sys
import math
from functools import reduce
import functools
import itertools
from collections import deque,Counter
from operator import mul
import copy
# ! /usr/bin/env python
# -*- coding: utf-8 -*-
import heapq
sys.setrecursionlimit(10**6)
INF =  float("inf")
import bisect

## a, bを無向辺として，隣接リストを作成
N = int(input())
a = [0] + list(map(int, input().split()))
al = [[] for i in range(N+1)]

for i in range(N-1):
    c, d = list(map(int, input().split()))
    al[c].append(d)
    al[d].append(c)

visited = [0 for i in range(N+1)]

LIS = [a[1]]
ans = [0 for i in range(N+1)]
ans[1] = 1

def dfs_rec(u):
    x = a[u]
    if x > LIS[-1]:
        pre_idx = len(LIS)
        pre_v = INF
        LIS.append(x)
    else:
        pre_idx = bisect.bisect_left(LIS, x)
        pre_v = LIS[pre_idx]
        LIS[bisect.bisect_left(LIS, x)] = x

    ans[u] = len(LIS)

    visited[u] = 1
    for v in al[u]:
        if visited[v] == 0:
            dfs_rec(v)

    if pre_v == INF:
        LIS.pop()
    else:
        LIS[pre_idx] = pre_v

dfs_rec(1)

for i in range(1,N+1):
    print(ans[i])



