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
sys.setrecursionlimit(200000)
from collections import deque
from bisect import bisect_left

N = int(input())
A = list(map(int, input().split()))
adj = [[] for i in range(N)]
for i in range(N - 1):
	u, v = map(int, input().split())
	adj[u - 1].append(v - 1)
	adj[v - 1].append(u - 1)


INF = 10 ** 10
done = [False] * N
done[0] = True
lisdp = [INF] * N
lisdp[0] = A[0]
change = [[-1, INF] for i in range(N)] #index, original
change[0] = [0, INF]
lisl = [0] * N
lisl[0] = 1

def dfs(v):
	for nv in adj[v]:
		if done[nv]:
			continue
		done[nv] = True
		ind = bisect_left(lisdp, A[nv])
		ori = lisdp[ind]
		change[nv] = [ind, ori]
		if ori == INF:
			lisl[nv] = lisl[v] + 1
		else:
			lisl[nv] = lisl[v]
		lisdp[ind] = A[nv]
		dfs(nv)
	lisdp[change[v][0]] = change[v][1]

dfs(0)
for i in range(N):
	print(lisl[i])	