  John Tromp introduced the so-called 'binary lambda calculus' as a way to
encode lambda terms in terms of 0-1-strings using the de Bruijn representation
along with a weighting scheme. 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 $n$ and according to Tromp's weights) 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.
  The methodology used is setting up the generating functions for the classes
of lambda terms. These are infinitely nested radicals which are investigated
then by a singularity analysis.
  We show further how some properties of random lambda terms can be analyzed
and present a way to sample lambda terms uniformly at random in a very
efficient way. This allows to generate terms of size more than one million
within a reasonable time, which is significantly better than the samplers
presented in the literature so far.
