I define a collections of nodes in trees which are called "Sweep-Covers" for their 'covering' of all the nodes in the tree by some ancestor-descendent relationship. Then, I analyze an algorithm for finding all sweep covers of a given size in any tree. The complexity of the algorithm is analyzed on a class of infinite $\Delta$-ary trees with constant path lengths between the $\Delta$-star internal nodes. The upper bound on the enumeration is analyzed with respect to maximum out-degree $\Delta$, size of sweep cover $n$, and path length $\gamma$. I prove that the Raney numbers are a strict lower bound for enumerating sweep-covers on these infinite trees.

中文翻译：

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扫频覆盖的枚举及其与拉尼数的关系

我在树中定义了一组节点，称为“Sweep-Covers”，因为它们通过某种祖先-后代关系“覆盖”了树中的所有节点。然后，我分析了一种算法，用于在任何树中查找给定大小的所有扫描覆盖。算法的复杂度是在一类无限$\Delta$-ary 树上分析的，其中$\Delta$-star 内部节点之间的路径长度是恒定的。枚举的上限根据最大出度 $\Delta$、扫描覆盖的大小 $n$ 和路径长度 $\gamma$ 进行分析。我证明了 Raney 数是枚举这些无限树上的扫掠覆盖的严格下界。