  John Tromp introduced the so-called 'binary lambda calculus' as a way to
encode lambda terms in terms of binary words. Later, Grygiel and Lescanne
conjectured that the number of binary lambda terms with $m$ free indices and of
size $n$ (encoded as binary words of length $n$) is $o(n^{-3/2} \tau^{-n})$ for
$\tau \approx 1.963448\ldots$. We generalize the proposed notion of size and
show that for several classes of lambda terms, including binary lambda terms
with $m$ free indices, the number of terms of size $n$ is $\Theta(n^{-3/2}
\rho^{-n})$ with some class dependent constant $\rho$, which in particular
disproves the above mentioned conjecture. A way to obtain lower and upper
bounds for the constant near the leading term is presented and numerical
results for a few previously introduced classes of lambda terms are given.
