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

inputFormat

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}

outputFormat

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

样例

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

1
2
3
3
4
4
5
2
2
3

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