Asymptotics of relaxed $k$-ary trees
<|reference_start|>Asymptotics of relaxed $k$-ary trees: 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.<|reference_end|>
@article{dastidar2024asymptotics,
    title={Asymptotics of relaxed $k$-ary trees},
    author={Manosij Ghosh Dastidar and Michael Wallner},
    journal={arXiv preprint arXiv:2404.08415},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.08415},
    primaryClass={math.CO cs.DM cs.DS}
}