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
#!/usr/bin/env python3
# Generated by 1.1.6 https://github.com/kyuridenamida/atcoder-tools
import sys
import bisect

INF = 1145141919

a = []
dp = []
ans = []
T = []


def dfs(v: int, p: int):
    index = bisect.bisect_left(dp, a[v])
    dp[index], a[v] = a[v], dp[index]

    ans[v] = bisect.bisect_left(dp, INF)

    for u in T[v]:
        if u != p:
            dfs(u, v)

    dp[index], a[v] = a[v], dp[index]


def solve(N: int, tmp_a: "List[int]", tmp_T: "List[List[int]]"):
    sys.setrecursionlimit(10 ** 6)

    global dp, ans, a, T
    dp = [INF] * N
    ans = [0] * N
    a = tmp_a
    T = tmp_T

    dfs(0, -1)

    print("\n".join(map(str, ans)))


def main():
    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    T = [[] for _ in range(N)]  # type: "List[List[int]]"
    for i in range(N - 1):
        u = int(next(tokens)) - 1
        v = int(next(tokens)) - 1
        T[u].append(v)
        T[v].append(u)
    solve(N, a, T)


if __name__ == '__main__':
    main()
