CF1517F.Reunion

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 $n$.

据报道，2050大会将于4月23日至25日在杭州云栖小镇举行，活动包括主题论坛、晨跑、露营等。

2050大会的 $n$ 名志愿者之间的关系可用一棵树（即一个含 $n$ 个顶点和 $n-1$ 条边的连通无向图）来表示。该树的 $n$ 个顶点对应 $n$ 名志愿者，编号为 $1,2,\ldots,n$。

我们定义两名志愿者 $i$ 和 $j$ 之间的距离 $\text{dis}(i,j)$ 为树上从顶点 $i$ 到顶点 $j$ 的最短路径所含的边数。当 $i=j$ 时，$\text{dis}(i,j)=0$。

部分志愿者可参加线下聚会，其余则不能。若对某名志愿者 $x$ 及某个非负整数 $r$，所有到 $x$ 的距离不超过 $r$ 的志愿者均能参加线下聚会，则可举办一个半径为 $r$ 的论坛。线下聚会的等级定义为所有可能举办的论坛中最大的半径值。

假设每名志愿者独立地以概率 $\frac{1}{2}$ 参加线下聚会。输出线下聚会等级的期望值。当无人能参加时，等级定义为 $-1$；当所有志愿者均能参加时，等级定义为 $n$。

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 $b$.

第一行包含一个整数 $n$（$2\le n\le 300$），表示志愿者的数量。

接下来的 $n-1$ 行每行包含两个整数 $a$ 和 $b$，表示顶点 $a$ 与顶点 $b$ 之间的一条边。

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 $0 \le x \lt M$ and $x \cdot q \equiv p \pmod{M}$.

输出期望等级对 $998\,244\,353$ 取模的结果。

形式化地，令 $M = 998\,244\,353$。可以证明答案可表示为一个既约分数 $\frac{p}{q}$，其中 $p$ 和 $q$ 均为整数，且 $q \not \equiv 0 \pmod{M}$。请输出满足 $p \cdot q^{-1} \bmod M$ 的整数。换言之，请输出满足 $0 \le x < M$ 且 $x \cdot q \equiv p \pmod{M}$ 的整数 $x$。

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 $\frac{3+1+1+(-1)}{2^3}=\frac{1}{2}$.

对于第一个例子，下表展示了所有可能的结果。“yes”表示志愿者可以参加线下聚会，“no”表示他无法参加。

$$\begin{array}{cccc} 1 & 2 & 3 & level\\\\ yes & yes & yes & 3\\\\ yes & yes & no & 1\\\\ yes & no & yes & 0\\\\ yes & no & no & 0\\\\ no & yes & yes & 1\\\\ no & yes & no & 0\\\\ no & no & yes & 0\\\\ no & no & no & -1\\\\ \end{array}$$

期望等级为 $\frac{3+1+1+(-1)}{2^3}=\frac{1}{2}$。