No abstract

We determine the limit of the expected value and the variance of the protection number of the root in simply generated trees, in P?lya trees, and in unlabelled non-plane binary trees, when the number of vertices tends to infinity. Moreover, we compute expectation and variance of the protection number of a randomly chosen vertex in all those tree classes. We obtain exact formulas as sum representations, where the obtained sums are rapidly converging thus allowing an efficient numerical computation of high accuracy.

The number of embeddings of a partially ordered set S in a partially ordered set T is the number of subposets of T isomorphic to S. If both, S and T , have only one unique maximal element, we define good embeddings as those in which the maximal elements of S and T overlap. We investigate the number of good and all embeddings of a rooted poset S in the family of all binary trees on n elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset S in the family of all planted plane trees of size n. We derive the asymptotic behaviour of good and all embeddings in all cases and we prove that the ratio of good embeddings to all is of the order Θ(1/ √ n) in all cases, where we provide the exact constants. Furthermore, we show that this ratio is non-decreasing with S in the plane binary case and asymptotically non-decreasing with S in the non-plane binary case and in the planted plane case. Finally, we comment on the case when S is disconnected.

No abstract

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