Random Forest Rank
Description
Let's define rank of undirected graph as rank of its adjacency matrix in $\mathbb{R}^{n \times n}$.
Given a tree. Each edge of this tree will be deleted with probability $1/2$, all these deletions are independent. Let $E$ be the expected rank of resulting forest. Find $E \cdot 2^{n-1}$ modulo $998244353$ (it is easy to show that $E \cdot 2^{n-1}$ is an integer).
First line of input contains $n$ ($1 \le n \le 5 \cdot 10^{5}$) — number of vertices.
Next $n-1$ lines contains two integers $u$ $v$ ($1 \le u, \,\, v \le n; \,\, u \ne v$) — indices of vertices connected by edge.
It is guaranteed that given graph is a tree.
Print one integer — answer to the problem.
Input
First line of input contains $n$ ($1 \le n \le 5 \cdot 10^{5}$) — number of vertices.
Next $n-1$ lines contains two integers $u$ $v$ ($1 \le u, \,\, v \le n; \,\, u \ne v$) — indices of vertices connected by edge.
It is guaranteed that given graph is a tree.
Output
Print one integer — answer to the problem.
Samples
3
1 2
2 3
6
4
1 2
1 3
1 4
14
4
1 2
2 3
3 4
18