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33
Transfer Theorems and Asymptotic Distributional Results for mary Search Trees
, 2004
"... We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary 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 soca ..."
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We derive asymptotics of moments and identify limiting distributions, under the random permutation model on mary 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 socalled 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 nonnormal 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 nonnormal distributions for all values of m.
A Hybrid of Darboux’s Method and Singularity Analysis in Combinatorial Asymptotics
 N O 1, JUNE 2006, R103, HTTP://WWW.COMBINATORICS.ORG/VOLUME_13/PDF/V13I1R103.PDF. ALGO 11
, 2006
"... 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 s ..."
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Cited by 7 (1 self)
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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.
Limit laws for functions of fringe trees for binary search trees and recursive trees
 In preparation
"... We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size k = kn in the binary search tree and the random recursive tree (of total size n) asymptotically ha ..."
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Cited by 6 (4 self)
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We prove limit theorems for sums of functions of subtrees of binary search trees and random recursive trees. In particular, we give simple new proofs of the fact that the number of fringe trees of size k = kn in the binary search tree and the random recursive tree (of total size n) asymptotically has a Poisson distribution if k →∞, and that the distribution is asymptotically normal for k = o( n). Furthermore, we prove similar results for the number of subtrees of size k with some required property P, for example the number of copies of a certain fixed subtree T. Using the Cramér–Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. As an application of the general results, we obtain a normal limit law for the number of `protected nodes in a binary search tree or random recursive tree. The proofs use a new version of a representation by Devroye, and Stein’s method (for both normal and Poisson approximation) together with certain couplings.
MULTIPLE ISOLATION OF NODES IN RECURSIVE TREES
"... ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a size ..."
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ABSTRACT. We introduce the problem of isolating several nodes in random recursive trees by successively removing random edges, and study the number of random cuts that are necessary for the isolation. In particular, we analyze the number of random cuts required to isolate ℓ selected nodes in a sizen random recursive tree for three different selection rules, namely (i) isolating all of the nodes labelled 1, 2..., ℓ (thus nodes located close to the root of the tree), (ii) isolating all of the nodes labelled n + 1 − ℓ, n + 2 − ℓ,... n (thus nodes located at the fringe of the tree), and (iii) isolating ℓ nodes in the tree, which are selected at random before starting the edgeremoval procedure. Using a generating functions approach we determine for these selection rules the limiting distribution behaviour of the number of cuts to isolate all selected nodes, for ℓ fixed and n → ∞. 1.
Limit Laws for the Number of Groups formed by Social Animals under the Extra Clustering Model
, 2014
"... We provide a complete description of the limiting behaviour of the number Xn of groups that are formed by social animals when the number n of animals tends to infinity. The analysis is based on a random model by Durand, Blum and François, where it is assumed that groups are formed more likely by an ..."
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We provide a complete description of the limiting behaviour of the number Xn of groups that are formed by social animals when the number n of animals tends to infinity. The analysis is based on a random model by Durand, Blum and François, where it is assumed that groups are formed more likely by animals which are genetically related. The random variable Xn can be described by a stochastic recurrence equation that is very similar to equations that occur in the stochastic analysis of divideandconquer algorithms although it does not fall into already known cases. In particular, we obtain (in the most interesting) “neutral model ” a curious central limit theorem, where the normalizing factor is√ Var(Xn)/2. In the nonneutral (or extra clustering) cases the results are completely different. We obtain either a mixture of a discrete and a continuous limit law or just a discrete limit law. 1
RNA pseudoknot structures with arclength ≥ 3 and stacklength ≥ σ. submitted
"... Abstract. In this paper we enumerate knoncrossing RNA pseudoknot structures with given minimum arc and stacklength. That is, we study the numbers of RNA pseudoknot structures with arclength ≥ 3, stacklength ≥ σ and in which there are at most k − 1 mutually crossing bonds, denoted by T [3] k,σ ( ..."
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Abstract. In this paper we enumerate knoncrossing RNA pseudoknot structures with given minimum arc and stacklength. That is, we study the numbers of RNA pseudoknot structures with arclength ≥ 3, stacklength ≥ 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 5noncrossing RNA structures with arclength ≥ 3 and stacklength ≥ 2 satisfy T [3] 3,2 (n) ∼ K3 n−52.5723n,
DESTRUCTION OF VERY SIMPLE TREES
"... Abstract. We consider the total cost of cutting down a random rooted tree chosen from a family of socalled very simple trees (which include ordered trees, dary 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 t ..."
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Abstract. We consider the total cost of cutting down a random rooted tree chosen from a family of socalled very simple trees (which include ordered trees, dary 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 onesided variant of the process the component not containing the root is discarded, whereas in the twosided 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 familyspecific manner) is that they only depend on α. 1.
ISOLATING NODES IN RECURSIVE TREES
"... Abstract. We consider the number of random cuts that are necessary to isolate the node with label λ, 1 ≤ λ ≤ n, in a random recursive tree of size n. At each stage of the edgeremoval procedure considered an edge is chosen at random from the tree and cut, separating the tree into two subtrees. The p ..."
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Abstract. We consider the number of random cuts that are necessary to isolate the node with label λ, 1 ≤ λ ≤ n, in a random recursive tree of size n. At each stage of the edgeremoval procedure considered an edge is chosen at random from the tree and cut, separating the tree into two subtrees. The procedure is then continued with the subtree containing the specified label λ, whereas the other subtree is discarded. The procedure stops when the node with label λ is isolated. Using a recursive approach we are able to give asymptotic expansions for all ordinary moments of the random variable Xn,λ, which counts the number of random cuts required to isolate the vertex with label λ in a random sizen recursive tree, for small labels, i. e., λ = l, and large labels, i. e., λ = n + 1 − l, with l ≥ 1 fixed and n → ∞. Moreover, we can characterize the limiting distribution of a scaled variant of Xn,λ, for the instance of large labels. 1.
A REPERTOIRE FOR ADDITIVE FUNCTIONALS OF UNIFORMLY DISTRIBUTED mARY SEARCH TREES
"... Abstract. Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on mary 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 to ..."
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Abstract. Using recent results on singularity analysis for Hadamard products of generating functions, we obtain the limiting distributions for additive functionals on mary 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 som−1 called shape functional; and (iii) 1n=m−1, which corresponds to the number of leaves. 1.
Central and local limit theorems for RNA structures
 J. Theor. Biol
, 2007
"... Abstract. A knoncrossing RNA pseudoknot structure is a graph over {1,..., n} without 1arcs, i.e. arcs of the form (i, i + 1) and in which there exists no kset of mutually intersecting arcs. In particular, RNA secondary structures are 2noncrossing RNA structures. In this paper we prove a central ..."
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Abstract. A knoncrossing RNA pseudoknot structure is a graph over {1,..., n} without 1arcs, i.e. arcs of the form (i, i + 1) and in which there exists no kset of mutually intersecting arcs. In particular, RNA secondary structures are 2noncrossing RNA structures. In this paper we prove a central and a local limit theorem for the distribution of the numbers of 3noncrossing RNA structures over n nucleotides with exactly h bonds. We will build on the results of [?] and [?], where the generating function of knoncrossing RNA pseudoknot structures and the asymptotics for its coefficients have been derived. The results of this paper explain the findings on the numbers of arcs of RNA secondary structures obtained by molecular folding algorithms and predict the distributions for knoncrossing RNA folding algorithms which are currently being developed. 1.