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Article Dans Une Revue Journal of Combinatorial Theory, Series A Année : 2021

Compacted binary trees admit a stretched exponential

Résumé

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 like where 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.
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hal-03132172 , version 1 (26-08-2022)

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Andrew Elvey Price, Wenjie Fang, Michael Wallner. Compacted binary trees admit a stretched exponential. Journal of Combinatorial Theory, Series A, 2021, 177, pp.105306. ⟨10.1016/j.jcta.2020.105306⟩. ⟨hal-03132172⟩
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