On the number of unary-binary tree-like structures with restrictions on the unary height

TL;DR

This paper investigates the enumeration of restricted unary-binary tree-like structures, including Motzkin trees and lambda-terms, using generating functions and singularity analysis, revealing unexpected phenomena and challenges in random generation methods.

Contribution

It introduces new asymptotic enumeration results for classes of Motzkin trees and lambda-terms with restrictions on unary nodes and nesting levels, highlighting complex generating functions.

Findings

Derived asymptotic counts for restricted Motzkin trees and lambda-terms.

Discovered unexpected behaviors in the generating functions involving nested square roots.

Identified difficulties in applying Boltzmann samplers for random generation of lambda-terms.

Abstract

We consider 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.