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 collections import deque
import bisect
import sys

sys.setrecursionlimit(200000)

N = int(input())
a = list(map(int, input().split()))
uvs = [list(map(int, input().split())) for _ in range(N-1)]
shortest = [10e+10 for _ in range(N)]
shortest[0] = 0

routes = [[] for _ in range(N)]

for uv in uvs:
    routes[uv[0]-1].append(uv[1]-1)
    routes[uv[1]-1].append(uv[0]-1)

lis = [10e+100 for _ in range(N)]
minimum = [10e+100 for _ in range(N)]
seen = [False for _ in range(N)]
seen[0] = True

def function(checking):
    insert_posi = bisect.bisect_left(lis, a[checking])
    preserved = lis[insert_posi]
    lis[insert_posi] = a[checking]
    minimum[checking] = bisect.bisect_left(lis, 10e+50)
    for route in routes[checking]:
        if seen[route] == False:
            seen[route] = True
            shortest[route] = shortest[checking]+1
            function(route)
    lis[insert_posi] = preserved
    return
        
function(0)

[print(x) for x in minimum]