F. Reunion
combinatorics; dp; trees
It is reported that the 2050 Conference will be held in Yunqi Town in Hangzhou from April 23 to 25, including theme forums, morning jogging, camping and so on.The relationship between the \(n\) volunteers of the 2050 Conference can be represented by a tree (a connected undirected graph with \(n\) vertices and \(n-1\) edges). The \(n\) vertices of the tree corresponds to the \(n\) volunteers and are numbered by \(1,2,\ldots, n\).We define the distance between two volunteers \(i\) and \(j\), dis\((i,j)\) as the number of edges on the shortest path from vertex \(i\) to vertex \(j\) on the tree. dis\((i,j)=0\) whenever \(i=j\).Some of the volunteers can attend the on-site reunion while others cannot. If for some volunteer \(x\) and nonnegative integer \(r\), all volunteers whose distance to \(x\) is no more than \(r\) can attend the on-site reunion, a forum with radius \(r\) can take place. The level of the on-site reunion is defined as the maximum possible radius of any forum that can take place.Assume that each volunteer can attend the on-site reunion with probability \(\frac{1}{2}\) and these events are independent. Output the expected level of the on-site reunion. When no volunteer can attend, the level is defined as \(-1\). When all volunteers can attend, the level is defined as \(n\).
The first line contains a single integer \(n\) (\(2\le n\le 300\)) denoting the number of volunteers.Each of the next \(n-1\) lines contains two integers \(a\) and \(b\) denoting an edge between vertex \(a\) and vertex \(b\).
Output the expected level modulo \(998\,244\,353\).Formally, let \(M = 998\,244\,353\). It can be shown that the answer can be expressed as an irreducible fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \not \equiv 0 \pmod{M}\). Output the integer equal to \(p \cdot q^{-1} \bmod M\). In other words, output such an integer \(x\) that \(0 \le x < M\) and \(x \cdot q \equiv p \pmod{M}\).
For the first example, the following table shows all possible outcomes. \(yes\) means the volunteer can attend the on-site reunion and \(no\) means he cannot attend. $$$\(\begin{array}{cccc} 1 & 2 & 3 & level\\ yes & yes & yes & 3\\ yes & yes & no & 1\\ yes & no & yes & 0\\ yes & no & no & 0\\ no & yes & yes & 1\\ no & yes & no & 0\\ no & no & yes & 0\\ no & no & no & -1\\ \end{array}\)\( The expected level is \)\frac{3+1+1+(-1)}{2^3}=\frac{1}{2}$$$.