CF1499F.Diameter Cuts

You are given an integer $k$ and an undirected tree, consisting of $n$ vertices.

The length of a simple path (a path in which each vertex appears at most once) between some pair of vertices is the number of edges in this path. A diameter of a tree is the maximum length of a simple path between all pairs of vertices of this tree.

You are about to remove a set of edges from the tree. The tree splits into multiple smaller trees when the edges are removed. The set of edges is valid if all the resulting trees have diameter less than or equal to $k$.

Two sets of edges are different if there is an edge such that it appears in only one of the sets.

Count the number of valid sets of edges modulo $998\,244\,353$.

给你一个整数 $k$ 和一棵包含 $n$ 个顶点的无向树。

某对顶点之间的一条简单路径（即每个顶点至多出现一次的路径）的长度定义为该路径所含边的数量。一棵树的直径是指其所有顶点对之间的简单路径的最大长度。

你将从该树中删除若干条边。删除边后，原树会分裂成若干棵更小的树。若删除后的所有子树的直径均不超过 $k$，则称所删除的边集是合法的。

若存在某条边仅属于两个边集中的一个，则称这两个边集不同。

请计算合法边集的总数，并对 $998\,244\,353$ 取模。

The first line contains two integers $n$ and $k$ ($2 \le n \le 5000$, $0 \le k \le n - 1$) — the number of vertices of the tree and the maximum allowed diameter, respectively.

Each of the next $n-1$ lines contains a description of an edge: two integers $v$ and $u$ ($1 \le v, u \le n$, $v \neq u$).

The given edges form a tree.

第一行包含两个整数 $n$ 和 $k$（$2 \le n \le 5000$，$0 \le k \le n - 1$），分别表示树的顶点数和允许的最大直径。

接下来的 $n-1$ 行每行描述一条边：两个整数 $v$ 和 $u$（$1 \le v, u \le n$，$v \neq u$）。

给定的边构成一棵树。

Print a single integer — the number of valid sets of edges modulo $998\,244\,353$.

输出一个整数——有效边集的数量对 $998\,244\,353$ 取模的结果。

In the first example the diameter of the given tree is already less than or equal to $k$. Thus, you can choose any set of edges to remove and the resulting trees will have diameter less than or equal to $k$. There are $2^3$ sets, including the empty one.

In the second example you have to remove the only edge. Otherwise, the diameter will be $1$, which is greater than $0$.

Here are the trees for the third and the fourth examples:

在第一个例子中，给定树的直径已经小于或等于 $k$。因此，你可以选择任意一组边进行删除，所得各子树的直径均小于或等于 $k$。这样的边集共有 $2^3$ 个（包含空集）。

在第二个例子中，你必须删除唯一的一条边；否则，所得树的直径为 $1$，大于 $0$。

第三个和第四个例子对应的树如下：