Patterns in random binary search trees

Abstract

In a randomly grown binary search tree BST of size n, any fixed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantified by a Gaussian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type of property otherwise known for strings and combinatorial tree models. They apply to paged trees or to quicksort with halting on short subfiles. As a consequence, various pointer saving strategies for maintaining trees obeying the random BST model can be precisely quantified. The methods used are based on analytic models, especially bivariate generating function subjected to singularity perturba-Ž .

Rare event asymptotics for a random walk in the quarter plane

“…The work on obtaining asymptotics from multivariate generating functions has been strongly motivated by recursively defined combinatorial structures like trees, see e.g. [9,12,13], and specific random walks or queueing models [4,14,15,19]. One of the central ideas in multivariate asymptotics is to exploit a functional equation to reduce multivariate problems to univariate contour integrals.…”

On Symmetries of Non-Plane Trees in a Non-Uniform Model

“…The non-plane trees in the binary search tree-like model were studied by analytic tools in [1,6] in a different context, and by a different method in [9]. In this work we use analytic combinatorics to initiate systematic studies of non-plane trees.…”

“…It should be noted that this recursion resembles a recurrence for binary search trees which was studied in [1,6]. For any t 1 , t 2 ∈ T n such that t 1 ∼ s and t 2 ∼ s it holds that P (T n = t 1 ) = P (T n = t 2 ).…”

“…In the first model to grow a tree built on n leaves we pick up randomly a leaf v, and then append two children to it, say v L and v R . Our second model (also known as the binary search tree model), is ubiquitous in the computer science literature, arising for example in the context of binary search trees formed by inserting a random permutation of [n − 1] into a binary search tree [1,6]. Under this model we generate a random tree T n as follows: at first, t is equal to the unique tree in T 1 and we associate a number n with its single vertex.…”

The Subtree Size Profile of Plane-oriented Recursive Trees

“…Also, as we will explain in more details below, results about the subtree size profile will in turn entail results about the occurrence of pattern sizes. Studying patterns in random trees is an important issue with many applications in computer science (for instance in the context of compressing; see Devroye [8] and Flajolet et al [19]) and mathematical biology (see [3] and Rosenberg [26]). …”