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We investigate the number of variables in two special subclasses oflambda-terms that are restricted by a bound of the number of abstractionsbetween a variable and its binding lambda, the so-called De-Bruijn index, or bya bound of the nesting levels of abstractions, \textit{i.e.}, the number of DeBruijn levels, respectively. These restrictions are on the one hand verynatural from a practical point of view, and on the other hand they simplify thecounting problem compared to that of unrestricted lambda-terms in such a waythat the common methods of analytic combinatorics are applicable.  We will show that the total number of variables is asymptotically normallydistributed for both subclasses of lambda-terms with mean and varianceasymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where theconstants $C$ and $\tilde{C}$ depend on the bound that has been imposed. So farwe just derived closed formulas for the constants in case of the class oflambda-terms with bounded De Bruijn index. However, for the other class oflambda-terms that we consider, namely lambda-terms with a bounded number of DeBruijn levels, we investigate the number of variables, as well as abstractionsand applications, in the different De Bruijn levels and thereby exhibit aso-called "unary profile" that attains a very interesting shape.

Distribution of variables in lambda-terms with restrictions on De Bruijn indices and De Bruijn levels