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John Tromp introduced the so-called 'binary lambda calculus' as a way toencode lambda terms in terms of 0-1-strings using the de Bruijn representationalong with a weighting scheme. Later, Grygiel and Lescanne conjectured that thenumber of binary lambda terms with $m$ free indices and of size $n$ (encoded asbinary 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 proposednotion of size and show that for several classes of lambda terms, includingbinary 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 $\rho$, whichin particular disproves the above mentioned conjecture.  The methodology used is setting up the generating functions for the classesof lambda terms. These are infinitely nested radicals which are investigatedthen by a singularity analysis.  We show further how some properties of random lambda terms can be analyzedand present a way to sample lambda terms uniformly at random in a veryefficient way. This allows to generate terms of size more than one millionwithin a reasonable time, which is significantly better than the samplerspresented in the literature so far.

Enumerating Lambda Terms by Weighted Length of Their De Bruijn Representation