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
#F
import sys
sys.setrecursionlimit(10**8)
from bisect import bisect_left, bisect_right
N = int(input())
a = list(map(int, input().split()))
lists=[[] for i in range(N)]
for i in range(N-1):
    u, v = map(int, input().split())
    lists[u-1].append(v-1)
    lists[v-1].append(u-1)

def search(v,LIS):
    if len(lists[v])==0:
        return
    for u in lists[v]:
        if check[u] == False:
            check[u] = True
            if a[u] > LIS[-1]:
                LIS.append(a[u])
                ans[u] = len(LIS)
                search(u,LIS)
                LIS.pop()
            else:
                ind = bisect_left(LIS,a[u])
                stack = LIS[ind]
                LIS[ind]=a[u]
                ans[u] = len(LIS)
                search(u,LIS)
                LIS[ind]=stack
    return

ans = [0]*N
check = [False]*N
check[0]=True
LIS = [a[0]]
ans[0]=1
search(0,LIS)
print(*ans, sep="\n")