Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

Abstract

We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio˛of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for˛2 OE0; 1 (with only convergence of finite moments when˛2 .1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for˛D 0 and a "quicksort type" limit law for˛D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

Profiles of random trees: correlation and width of random recursive trees and binary search trees

“…The distributional properties of both types of profile (X n,k and I n,k ) are similar to those of Y n,k ; see Fuchs et al (2005) for details.…”

“…Many properties of Y n,k are known. We briefly summarize the interesting phenomena exhibited by Y n,k , as follows; see Drmota and Hwang (2005) and Fuchs et al (2005) for more information.…”

“…Such arguments, based on considering the random polynomial k Y n,k z k , also provide a natural interpretation of the result (see Fuchs et al (2005)) that the sequence of random variables (Y n,k − E(Y n,k ))/var(Y n,k ) 1/2 converges to the same limit law as the total path length T n := k kX n,k when k ∼ log n and |k−log n| → ∞; see Section 3 for more details.…”

“…This paper is a sequel to Drmota and Hwang (2005) and Fuchs et al (2005), in which the authors addressed the limit distributions of profiles (the numbers of nodes at each level) in random recursive trees and binary search trees. Further to the many intriguing phenomena unveiled there, we show in this paper that the correlation coefficients of two level sizes, in both classes of tree, exhibit sharp sign-changes.…”

“…Recursive trees (following Meir and Moon (1974)) have also appeared in other fields under different names: 'concave node-weighted trees' in Tapia and Myers (1967), 'growing trees' in Na and Rapoport (1970), 'pyramid scheme' in Gastwirth (1977), 'heap-ordered trees' in Grossman and Larson (1989), and 'random circuits with fanin one' in Arya et al (1999). They have been introduced as simple growing models for several real-life networks like social systems (Na and Rapoport (1970)), sales-distribution networks (Moon (1974)), and the Internet; see Fuchs et al (2005) for more references. Their simple tree representations also found applications in many linear tree algorithms; see Mitchell et al (1979).…”