On the subtree size profile of binary search trees

For random trees T generated by the binary search tree algorithm from uniformly distributed input we consider the subtree size profile, which maps k ∈ ℕ to the number of nodes in T that root a subtree of size k. Complementing earlier work by Devroye, by Feng, Mahmoud and Panholzer, and by Fuchs, we obtain results for the range of small k-values and the range of k-values proportional to the size n of T. In both cases emphasis is on the process view, i.e., the joint distributions for several k-values. We also show that the dynamics of the tree sequence lead to a qualitative difference between the asymptotic behaviour of the lower and the upper end of the profile. Copyright © 2010 Cambridge University Press

Persisting randomness in randomly growing discrete structures: graphs and search trees

The successive discrete structures generated by a sequential algorithm from random input constitute a Markov chain that may exhibit long term dependence on its first few input values. Using examples from random graph theory and search algorithms we show how such persistence of randomness can be detected and quantified with techniques from discrete potential theory. We also show that this approach can be used to obtain strong limit theorems in cases where previously only distributional convergence was known.Comment: Official journal fil

Normal Limiting Distribution of the Size of Binary Interval Trees

The limiting distribution of the size of binary interval tree is investigated. Our illustration is based on the contraction method, and it is quite different from the case in one-sided binary interval tree. First, we build a distributional recursive equation of the size. Then, we draw the expectation, the variance, and some high order moments. Finally, it is shown that the size (with suitable standardization) approaches the standard normal random variable in the Zolotarev metric space

Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees

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 $k=k_n$ in the binary search tree and the random recursive tree (of total size $n$) asymptotically has a Poisson distribution if $k\rightarrow\infty$, and that the distribution is asymptotically normal for $k=o(\sqrt{n})$. 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 Cram\'er-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 $\ell$-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

Trickle-down processes and their boundaries

It is possible to represent each of a number of Markov chains as an evolving sequence of connected subsets of a directed acyclic graph that grow in the following way: initially, all vertices of the graph are unoccupied, particles are fed in one-by-one at a distinguished source vertex, successive particles proceed along directed edges according to an appropriate stochastic mechanism, and each particle comes to rest once it encounters an unoccupied vertex. Examples include the binary and digital search tree processes, the random recursive tree process and generalizations of it arising from nested instances of Pitman's two-parameter Chinese restaurant process, tree-growth models associated with Mallows' phi model of random permutations and with Schuetzenberger's non-commutative q-binomial theorem, and a construction due to Luczak and Winkler that grows uniform random binary trees in a Markovian manner. We introduce a framework that encompasses such Markov chains, and we characterize their asymptotic behavior by analyzing in detail their Doob-Martin compactifications, Poisson boundaries and tail sigma-fields.Comment: 62 pages, 8 figures, revised to address referee's comment