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,math,copy,queue,itertools,bisect,collections,heapq
LI = lambda : [int(x) for x in sys.stdin.readline().split()]
NI = lambda : int(sys.stdin.readline())
INF = 10**9+1

N = NI()
a = LI()

root = [[] for _ in range(N)]
for _ in range(N-1):
    u,v = LI()
    root[u-1].append(v-1)
    root[v-1].append(u-1)

lis = [INF for _ in range(N)]
q = []

q.append((0,0,0))
ans = [-1 for _ in range(N)]
while q:
    c,p,v = q.pop()
    if c == 0:
        x = bisect.bisect_left(lis,a[p])
        y = lis[x]
        if y == INF: v +=1
        if y > a[p]: lis[x] = a[p]
        ans[p] = v
        q.append((1,x,y))
        for d in root[p]:
            if ans[d] < 0: q.append((0,d,v))
    else:
        lis[p] = v
print(*ans,sep='\n')