  We consider combinatorial aspects of $\lambda$-terms in the model based on de
Bruijn indices where each building constructor is of size one. Surprisingly,
the counting sequence for $\lambda$-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 $\lambda$-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 $\lambda$-terms in neutral, normal, and head normal forms. Finally, we
investigate the asymptotic density of $\lambda$-terms containing arbitrary
fixed subterms showing that, inter alia, strongly normalising or typeable terms
are asymptotically negligible in the set of all $\lambda$-terms.
