Weighted height of random trees

We consider a model of random trees similar to the split trees of Devroye [30] in which a set of items is recursively partitioned. Our model allows for more flexibility in the choice of the partitioning procedure, and has weighted edges. We prove that for this model, the height Hn of a random tree is asymptotic to c log n in probability for a constant c that is uniquely characterized in terms of multivariate large deviations rate functions. This extension permits us to obtain the height of pebbled tries, pebbled ternary search tries, d-ary pyramids, and to study geometric properties of partitions generated by k-d trees. The model also includes all polynomial families of increasing trees recently studied by Broutin, Devroye, McLeish, and de la Salle [17]. Keywords and phrases: Height, random tree, split tree, branching process, large deviations, probabilistic analysis.

• Type: Journal
• Year: 2008
• Where: ACTA
• Authors: Nicolas Broutin, Luc Devroye, Erin McLeish
• Keywords: ACTA 2007, ACTA 2008, Large Deviations, Model, Random Tree