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
import sys
from collections import defaultdict
from bisect import bisect_left
sys.setrecursionlimit(1000000)
input=sys.stdin.readline


N = int(input())
*a, = map(int, input().split())
d = defaultdict(list)
for _ in range(N-1):
    u, v = map(int, input().split())
    d[u].append(v)
    d[v].append(u)

INF = 10**20
l = [INF]*(N)
q = []
ans = [0]*(N+1)

def dfs(s, root):
    v = bisect_left(l, a[s-1])
    q.append((v, l[v]))
    l[v] = a[s-1]
    ans[s] = bisect_left(l, INF)

    for t in d[s]:
        if t==root:continue
        dfs(t, s)
    
    ba, bb = q.pop()
    l[ba] = bb

dfs(1, 0)
for i in range(1, N+1):
    print(ans[i])
