Distribution of Variables in Lambda-Terms with Restrictions on De Bruijn Indices and De Bruijn Levels

We consider two special subclasses of lambda-terms that are restricted by a bound on the number of abstractions between a variable and its binding lambda, the so-called De-Bruijn index, or by a bound on the nesting levels of abstractions, i.e., the number of De Bruijn levels, respectively. We show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to $Cn$ and $\tilde{C}n$, respectively, where the constants $C$ and $\tilde{C}$ depend on the bound that has been imposed. For the class of lambda-terms with bounded De Bruijn index we derive closed formulas for the constant. For the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of De Bruijn levels, we show quantitative and distributional results on the number of variables, as well as abstractions and applications, in the different De Bruijn levels and thereby exhibit a so-called "unary profile" that attains a very interesting shape.  Our results give a combinatorial explanation of an earlier discovered strange phenomenon exhibited by the counting sequence of this particular class of lambda-terms.