CF1067E.Random Forest Rank

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.