p02698 AtCoder Beginner Contest 165 - LIS on Tree
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
```python
from bisect import bisect_left

n = int(input())
A = [0] + list(map(int, input().split()))
graph = [[] for _ in range(n + 1)]
for _ in range(n - 1):
    u, v = map(int, input().split())
    graph[v].append(u)
    graph[u].append(v)

start = 1

stack = [1]
par = [-1] * (n + 1)
ans = [0] * (n + 1)
used = [False] * (n + 1)
infi = 10 ** 20
LIS = [infi] * (n + 1)
LIS[0] = -infi
position = [(-1, -1)] * (n + 1)


def hantei(val, L):
    pos = bisect_left(L, val)
    pre = L[pos]
    L[pos] = val
    cnt = bisect_left(L, infi)
    return L, pos, pre, cnt


while stack:
    v = stack[-1]
    if not used[v]:
        LIS, pos, pre, cnt = hantei(A[v], LIS)
        position[v] = (pos, pre)
        ans[v] = cnt - 1
    used[v] = True
    if not graph[v]:
        _ = stack.pop()
        basho, atai = position[v]
        LIS[basho] = atai
        continue
    u = graph[v].pop()
    if u == par[v]:
        continue
    par[u] = v
    stack.append(u)

print(*ans[1:], sep="\n")

```