A relaxed $k$-ary tree is an ordered directed acyclic graph with a unique source and sink in which every node has out-degree $k$. These objects arise in the compression of trees in which some repeated subtrees are factored and repeated appearances are replaced by pointers. We prove an asymptotic theta-result for the number of relaxed $k$-ary tree with $n$ nodes for $n \to \infty$. This generalizes the previously proved binary case to arbitrary finite arity, and shows that the seldom observed phenomenon of a stretched exponential term $e^{c n^{1/3}}$ appears in all these cases. We also derive the recurrences for compacted $k$-ary trees in which all subtrees are unique and minimal deterministic finite automata accepting a finite language over a finite alphabet.

翻译：松弛k叉树是一种有序有向无环图，具有唯一的源点和汇点，其中每个节点的出度为k。这类结构出现在树的压缩中，其中一些重复的子树被分解，重复出现被指针替代。我们证明了当n趋于无穷时，具有n个节点的松弛k叉树数量的渐近θ结果。这一结果将先前证明的二叉情形推广到任意有限出度，并表明一种罕见的拉伸指数项e^{c n^{1/3}}现象在所有情形中均出现。我们还导出了紧凑k叉树的递推关系，其中所有子树都是唯一的，并研究了接受有限字母表上有限语言的最小确定型有限自动机。