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
from bisect import bisect_left
def main():
    sys.setrecursionlimit(202020)
    N = int(input())
    A = [0] + list(map(int, input().split()))
    G = [[] for _ in range(N+1)]
    for _ in range(N-1):
        u, v = map(int, input().split())
        G[u].append(v)
        G[v].append(u)
    L = [A[1]]
    Ans = [0] * (N+1)
    def dfs(v):
        Ans[v] = len(L)
        for u in G[v]:
            G[u].remove(v)
            Au = A[u]
            if Au > L[-1]:
                L.append(Au)
                dfs(u)
                del L[-1]
            else:
                idx = bisect_left(L, Au)
                old = L[idx]
                L[idx] = Au
                dfs(u)
                L[idx] = old
    dfs(1)
    print("\n".join(map(str, Ans[1:])))

main()
