Compacted binary trees admit a stretched exponential

A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likeΘ(n!4ne3a1n1/3n3/4), where a1≈−2.338 is the larg...

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Bibliographic Details
Published inJournal of combinatorial theory. Series A Vol. 177; p. 105306
Main Authors Elvey Price, Andrew, Fang, Wenjie, Wallner, Michael
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2021
Elsevier
Subjects
Airy function
Asymptotics
Bijection
Combinatorics
Compacted trees
Directed acyclic graphs
Dyck paths
Finite languages
Mathematics
Minimal automata
Stretched exponential
Stretched exponential
Bijection
Compacted trees
Airy function
Directed acyclic graphs
Minimal automata
Dyck paths
Finite languages
Asymptotics
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Summary:A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likeΘ(n!4ne3a1n1/3n3/4), where a1≈−2.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity for computing the number of compacted trees up to a given size. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.
ISSN:0097-3165
1096-0899
DOI:10.1016/j.jcta.2020.105306