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 bisect import bisect_left

N = int(input())
A = list(map(int, input().split()))

edges = [[] for _ in range(N + 1)]
for _ in range(N - 1):
    fr, to = map(int, input().split())
    edges[fr].append(to)
    edges[to].append(fr)

order = []
parent = [-1] * (N + 1)
st = [1]
while st:
    now = st.pop()
    order.append(now)
    if now > 0:
        st.append(-now)
        for to in edges[now]:
            if to == parent[now]:
                continue
            st.append(to)
            parent[to] = now

L = [-10**18]
ans = [-1] * (N + 1)
st = [(-1, -1)] * (N + 1)

for now in order:
    if now < 0:
        val, idx = st[-now]
        if val == -1:
            L.pop()
        if val > 0:
            L[idx] = val
        continue

    a = A[now - 1]
    i = bisect_left(L, a)

    if len(L) == i:
        L.append(a)
    else:
        st[now] = (L[i], i)
        if L[i] > a:
            L[i] = a

    ans[now] = len(L) - 1

print(*ans[1:], sep='\n')
