We derive asymptotics of moments and identify limiting distributions, under the random permutation model on m-ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that (i) for small toll sequences (tn) [roughly, tn = O(n1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, tn = ω(n1/2) but tn = o(n)] we have convergence to non-normal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior if m ≥ m0 + 1; and (iii) for large toll sequences [roughly, tn = ω(n)] we have convergence to non-normal distributions for all values of m.

Abstract. A “hybrid method”, dedicated to asymptotic coefficient extraction in combinatorial generating functions, is presented, which combines Darboux’s method and singularity analysis theory. This hybrid method applies to functions that remain of moderate growth near the unit circle and satisfy suitable smoothness assumptions—this, even in the case when the unit circle is a natural boundary. A prime application is to coefficients of several types of infinite product generating functions, for which full asymptotic expansions (involving periodic fluctuations at higher orders) can be derived. Examples relative to permutations, trees, and polynomials over finite fields are treated in this way.

We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive integer k. We show the existence of a joint limit distribution for these parameters in the limit of infinite size, if the size generating function has a square root as dominant singularity. The limit distribution coincides with that of integrals of k-th powers of the standard Brownian excursion. Our approach yields a recursion for the moments of the limit distribution. It can be used to analyse asymptotic expansions of the moments, and it admits an extension to other types of singularity.

Abstract. In this paper we study k-noncrossing RNA structures with arc-length ≥ 3, i.e. RNA molecules in which for any i, the nucleotides labeled i and i + j (j = 1,2) cannot form a bond and in which there are at most k − 1 mutually crossing arcs. Let Sk,3(n) denote their number. Based on a novel functional equation for the generating function P n≥0 Sk,3(n)zn, we derive for arbitrary k ≥ 3 exponential growth factors and for k = 3 the subexponential factor. Our main result is the derivation of the formula S3,3(n) ∼

Abstract. In this paper we enumerate k-noncrossing RNA pseudoknot structures with given minimum arc- and stack-length. That is, we study the numbers of RNA pseudoknot structures with arc-length ≥ 3, stack-length ≥ σ and in which there are at most k − 1 mutually crossing bonds, denoted by T [3] k,σ (n). In particular we prove that the numbers of 3, 4 and 5-noncrossing RNA structures with arc-length ≥ 3 and stack-length ≥ 2 satisfy T [3] 3,2 (n) ∼ K3 n−52.5723n,

Search trees are fundamental data structures in computer science. We study functionals on random search trees that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so-called shape functional fall under this framework. Our goal is to derive asymptotics of moments and identify limiting distributions of these functionals under two commonly studied probability models—the random permutation model and the uniform model. For the random permutation model, our approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. For the uniform model, our approach is based on the complex-analytic tool of singularity analysis. To facilitate a systematic analysis of these additive functionals we extend singularity analysis, a class of methods by which one can translate on a term-by-term basis an asymptotic expansion of a functional around its dominant singularity into a corresponding expansion for the Taylor coefficients of the function. The most important extension is the determination of how singularities are composed under the operation of Hadamard product of analytic power series. The transfer theorems derived are used in conjunction with the method of moments to establish limit laws for m-ary search trees under the random permutation model. For the uniform model on binary search trees, the extended singularity analysis toolkit is employed to establish the asymptotic behavior of the moments of a wide class of functionals. These asymptotics are used, again in conjunction with the method of moments, to derive limit laws.

Abstract. We present a detailed study of left-right-imbalance measures for random binary search trees under the random permutation model, i.e., where binary search trees are generated by random permutations of {1, 2,..., n}. For random binary search trees of size n we study (i) the difference between the left and the right depth of a randomly chosen node, (ii) the difference between the left and the right depth of a specified node j = j(n), and (iii) the difference between the left and the right pathlength, and show for all three imbalance measures limiting distribution results. 1.

Abstract. Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on m-ary search trees on n keys with toll sequence (i) nα with α ≥ 0 (α = 0 and α = 1 correspond roughly to the space requirement and total path length, respectively); (ii) ln ` n ´, which corresponds to the so-m−1 called shape functional; and (iii) 1n=m−1, which corresponds to the number of leaves. 1.

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Abstract. We consider the total cost of cutting down a random rooted tree chosen from a family of so-called very simple trees (which include ordered trees, d-ary trees, and Cayley trees); these form a subfamily of simply generated trees. At each stage of the process an edge is chose at random from the tree and cut, separating the tree into two components. In the one-sided variant of the process the component not containing the root is discarded, whereas in the two-sided variant both components are kept. The process ends when no edges remain for cutting. The cost of cutting an edge from a tree of size n is assumed to be n α. Using singularity analysis and the method of moments, we derive the limiting distribution of the total cost accrued in both variants of this process. A salient feature of the limiting distributions obtained (after normalizing in a family-specific manner) is that they only depend on α. 1.