On the Number of Unary-Binary Tree-Like Structures with Restrictions on the Unary Height

On the Number of Unary-Binary Tree-Like Structures with Restrictions on the Unary Height We investigate various classes of Motzkin trees as well as lambda-terms for which we derive asymptotic enumeration results. These classes are defined through various restrictions concerning the unary nodes or abstractions, respectively: we either bound their number or the allowed levels of nesting. The enumeration is done by means of a generating function approach and singularity analysis. The generating functions are composed of nested square roots and exhibit unexpected phenomena in some of the cases. Furthermore, we present some observations obtained from generating such terms randomly and explain why usually powerful tools for random generation, such as Boltzmann samplers, face serious difficulties in generating lambda-terms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Combinatorics Springer Journals

On the Number of Unary-Binary Tree-Like Structures with Restrictions on the Unary Height

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Publisher
Springer International Publishing
Copyright
Copyright © 2018 by The Author(s)
Subject
Mathematics; Combinatorics
ISSN
0218-0006
eISSN
0219-3094
D.O.I.
10.1007/s00026-018-0371-7
Publisher site
See Article on Publisher Site

Abstract

We investigate various classes of Motzkin trees as well as lambda-terms for which we derive asymptotic enumeration results. These classes are defined through various restrictions concerning the unary nodes or abstractions, respectively: we either bound their number or the allowed levels of nesting. The enumeration is done by means of a generating function approach and singularity analysis. The generating functions are composed of nested square roots and exhibit unexpected phenomena in some of the cases. Furthermore, we present some observations obtained from generating such terms randomly and explain why usually powerful tools for random generation, such as Boltzmann samplers, face serious difficulties in generating lambda-terms.

Journal

Annals of CombinatoricsSpringer Journals

Published: Feb 13, 2018

References

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