Unary profile of lambda terms with restricted De Bruijn indices

In this paper we present an average-case analysis of closed lambda terms with restricted values De Bruijn indices in the model where each occurrence a variable contributes one to size. Given fixed integer k, term which all are bounded by k has following shape: It starts levels, forming so-called hat term, some number k-colored Motzkin trees attached. By means analytic combinatorics, show that size is constant on average and levels n asymptotically Θ(√ n). Combining these two facts, conclude maximal non-empty level restrictions is, average, also order √ n. On basis, provide unary profile such terms.