Combinatorics of $$\lambda$$-terms: a natural approach

We consider combinatorial aspects of λ-terms in the model based on de Bruijn indices where each building constructor is of size one. Surprisingly, the counting sequence for λ-terms corresponds also to two families of binary trees, namely black-white trees and zigzag-free ones. We provide a constructive proof of this fact by exhibiting appropriate bijections. Moreover, we identify the sequence of Motzkin numbers with the counting sequence for neutral λ-terms, giving a bijection which, in consequence, results in an exact-size sampler for the latter based on the exact-size sampler for Motzkin trees of Bodini et alli. Using the powerful theory of analytic combinatorics, we state several results concerning the asymptotic growth rate of λ-terms in neutral, normal, and head normal forms. Finally, we investigate the asymptotic density of λ-terms containing arbitrary fixed subterms showing that, inter alia, strongly normalising or typeable terms are asymptotically negligible in the set of all λ-terms.