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arXiv · 2404.08415

Asymptotics of relaxed $k$-ary trees

Manosij Ghosh Dastidar·Michael Wallner
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Abstract

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.

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Manosij Ghosh Dastidar, Michael Wallner. 2024-04-12. Asymptotics of relaxed $k$-ary trees. https://arxiv.org/abs/2404.08415

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