Problem Statement

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 $k$.