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
input = sys.stdin.readline
from collections import deque
import bisect

n = int(input())
a = list(map(int,input().split()))

links = [[] for _ in range(n+1)]
for _ in range(n-1):
    v,w = map(int, input().split())
    links[v].append(w)
    links[w].append(v)

inf = 10**10
lis = [inf] * n
inf_ind = 0
parent = [-1]*(n+1)
operate = [0] * (n+1)
ans = [-1]*(n+1)
d = deque()
d.append(1)
while(d):
    now = d.popleft()
    if(operate[now]==0):
        p = parent[now]
        if(p!=-1):
            d.appendleft(now)
        for child in links[now]:
            if(child != p):
                parent[child] = now
                d.appendleft(child)

        a_now = a[now-1]
        ind = bisect.bisect_left(lis,a_now)
        operate[now] = (ind,lis[ind],a_now)
        lis[ind] = a_now
        if(inf_ind == ind):
            inf_ind += 1
        ans[now] = inf_ind
    else:
        ind,before,after = operate[now]
        lis[ind] = before
        if(before == inf):
            inf_ind -= 1
    # print(lis)

for i in ans[1:]:
    print(i)