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
sys.setrecursionlimit(1000000)
INF = pow(10, 10)

changed = []
input = sys.stdin.readline
N = int(input())
A = [int(a) for a in input().split()]
Edge = [[] for _ in range(N)]
for _ in range(N-1):
    u, v = map(int, input().split())
    Edge[u-1].append(v-1)
    Edge[v-1].append(u-1)
LIS = [INF] * N

def dfs(i, pre, Ans):
    changeID = bisect_left(LIS, A[i])
    changed.append((changeID, LIS[changeID]))
    LIS[changeID] = min(A[i], LIS[changeID])
    #print(i, LIS)
    #print(changed)
    Ans[i] = bisect_left(LIS, INF) 
    for nextN in Edge[i]:
        if nextN != pre: dfs(nextN, i, Ans)
    backID, backV = changed.pop()
    LIS[backID] = backV
    return 0

def solve():
    Ans = [1] * N
    dfs(0, 0, Ans)
    print("\n".join(map(str, Ans)))
    
    return 0

if __name__ == "__main__":
    solve()