80 research outputs found
Limiting distributions for additive functionals on Catalan trees
Additive tree functionals represent the cost of many divide-and-conquer
algorithms. We derive the limiting distribution of the additive functionals
induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log
n (the so-called shape functional) on uniformly distributed binary search
trees, sometimes called Catalan trees. The Gaussian law obtained in the latter
case complements the central limit theorem for the shape functional under the
random permutation model. Our results give rise to an apparently new family of
distributions containing the Airy distribution (\alpha = 1) and the normal
distribution [case (b), and case (a) as ]. The main
theoretical tools employed are recent results relating asymptotics of the
generating functions of sequences to those of their Hadamard product, and the
method of moments.Comment: 30 pages, 4 figures. Version 2 adds background information on
singularity analysis and streamlines the presentatio
A repertoire for additive functionals of uniformly distributed m-ary search trees
Using recent results on singularity analysis for Hadamard products of
generating functions, we obtain the limiting distributions for additive
functionals on -ary search trees on keys with toll sequence (i)
with ( and correspond roughly
to the space requirement and total path length, respectively); (ii) , which corresponds to the so-called shape functional; and (iii)
, which corresponds to the number of leaves.Comment: 26 pages; v2 expands on the introduction by comparing the results
with other probability model
The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance
For two decades, the Colless index has been the most frequently used
statistic for assessing the balance of phylogenetic trees. In this article,
this statistic is studied under the Yule and uniform model of phylogenetic
trees. The main tool of analysis is a coupling argument with another well-known
index called the Sackin statistic. Asymptotics for the mean, variance and
covariance of these two statistics are obtained, as well as their limiting
joint distribution for large phylogenies. Under the Yule model, the limiting
distribution arises as a solution of a functional fixed point equation. Under
the uniform model, the limiting distribution is the Airy distribution. The
cornerstone of this study is the fact that the probabilistic models for
phylogenetic trees are strongly related to the random permutation and the
Catalan models for binary search trees.Comment: Published at http://dx.doi.org/10.1214/105051606000000547 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Destruction of very simple trees
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, -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 is
assumed to be . 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
.Comment: 20 pages; Version 2 corrects some minor error and fixes a few typo
Weak convergence of random p-mappings and the exploration process of inhomogeneous continuum random trees
We study the asymptotics of the -mapping model of random mappings on
as gets large, under a large class of asymptotic regimes for the underlying
distribution . We encode these random mappings in random walks which are
shown to converge to a functional of the exploration process of inhomogeneous
random trees, this exploration process being derived (Aldous-Miermont-Pitman
2003) from a bridge with exchangeable increments. Our setting generalizes
previous results by allowing a finite number of ``attracting points'' to
emerge.Comment: 16 page
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