The wiener index of simply generated random trees

Authors:
Svante Janson
Affiliations:
Department of Mathematics, Uppsala University, P.O. Box 480, S-751 06 Uppsala, Sweden
Venue:
Random Structures & Algorithms
Year:
2003

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Abstract

Asymptotics are obtained for the mean, variance, and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton-Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions.