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 bisect
I = [int(_) for _ in open(0).read().split()]
N = I[0]
A = [0] + I[1:1 + N]
U = [0] + I[1 + N::2]
V = [1] + I[2 + N::2]
G = [set() for _ in range(N + 1)]
for u, v in zip(U, V):
    G[u].add(v)
    G[v].add(u)
ans = [10**10] * (N + 1)
stack = [(0, 1, 1, A[1], 0)]
dp = [-10 ** 10] + [10 ** 10] * N
while stack:
    i, j, p, q, d = stack.pop()
    if d:
        dp[p] = q
        continue
    stack += [(j, i, p, dp[p], 1)]
    dp[p] = q
    ans[j] = bisect.bisect_left(dp, 10 ** 10) - 1
    while G[j]:
        k = G[j].pop()
        if i == k:
            continue
        q = A[k]
        p = bisect.bisect_left(dp, q)
        stack += [(j, k, p, q, 0)]
print(*ans[1:], sep='\n')
