Transfer Theorems and Asymptotic Distributional Results for m-ary Search Trees

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 $(t_n)$ [roughly, $t_n =O(n^{1 / 2})$] we have asymptotic normality if $m \leq 26$ and typically periodic behavior if $m \geq 27$; (ii) for moderate toll sequences [roughly, $t_n = \omega(n^{1 / 2})$ but $t_n = o(n)$] we have convergence to non-normal distributions if $m \leq m_0$ (where $m_0 \geq 26$) and typically periodic behavior if $m \geq m_0 + 1$; and (iii) for large toll sequences [roughly, $t_n = \omega(n)$] we have convergence to non-normal distributions for all values of m.Comment: 35 pages, 1 figure. Version 2 consists of expansion and rearragement of the introductory material to aid exposition and the shortening of Appendices A and B.

Congruence properties of depths in some random trees

Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n tends to infinty. The same is true for the number of vertices of depth divisible by m for m=3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m at least 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m=6 is intermediate: the number is asymptotically normal but the variance is of order n log n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and perhaps explain, this non-intuitive behavious using a translation to a generalized Polya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m at most 8 but not for m at least 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton-Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.Comment: 23 page

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 $m$-ary search trees on $n$ keys with toll sequence (i) $n^\alpha$ with $\alpha \geq 0$ ($\alpha=0$ and $\alpha=1$ correspond roughly to the space requirement and total path length, respectively); (ii) $\ln \binom{n}{m-1}$, which corresponds to the so-called shape functional; and (iii) $\mathbf{1}_{n=m-1}$, which corresponds to the number of leaves.Comment: 26 pages; v2 expands on the introduction by comparing the results with other probability model

Phase Transition in the Aldous-Shields Model of Growing Trees

We study analytically the late time statistics of the number of particles in a growing tree model introduced by Aldous and Shields. In this model, a cluster grows in continuous time on a binary Cayley tree, starting from the root, by absorbing new particles at the empty perimeter sites at a rate proportional to c^{-l} where c is a positive parameter and l is the distance of the perimeter site from the root. For c=1, this model corresponds to random binary search trees and for c=2 it corresponds to digital search trees in computer science. By introducing a backward Fokker-Planck approach, we calculate the mean and the variance of the number of particles at large times and show that the variance undergoes a `phase transition' at a critical value c=sqrt{2}. While for c>sqrt{2} the variance is proportional to the mean and the distribution is normal, for c<sqrt{2} the variance is anomalously large and the distribution is non-Gaussian due to the appearance of extreme fluctuations. The model is generalized to one where growth occurs on a tree with $m$ branches and, in this more general case, we show that the critical point occurs at c=sqrt{m}.Comment: Latex 17 pages, 6 figure

Asymptotic distribution of two-protected nodes in ternary search trees

We study protected nodes in $m$-ary search trees, by putting them in context of generalised P\'olya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m$-ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $m$; we conjecture that the method yields an asymptotically normal distribution for all $m\leq 26$. The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler P\'olya urn (that is similar to the one that has earlier been used to study the total number of nodes in $m$-ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $m\leq 26$