On the distribution of distances between specified nodes in increasing trees

Authors:
Markus Kuba;Alois Panholzer
Affiliations:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria;Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040 Wien, Austria
Venue:
Discrete Applied Mathematics
Year:
2010

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Abstract

We study the quantity distance between nodejand nodenin a random tree of sizen chosen from a family of increasing trees. For those subclass of increasing tree families, which can be constructed via a tree evolution process, we give closed formulae for the probability distribution, the expectation and the variance. Furthermore we derive a distributional decomposition of the random variable considered and we show a central limit theorem of this quantity, for arbitrary labels 1@?j~. Such tree models are of particular interest in applications, e.g., the widely used models of recursive trees, plane-oriented recursive trees and binary increasing trees are special instances and are thus covered by our results.