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In this paper, we investigate the degree profile and Gini index of randomcaterpillar trees (RCTs). We consider RCTs which evolve in two differentmanners: uniform and nonuniform. The degrees of the vertices on the centralpath (i.e., the degree profile) of a uniform RCT follow a multinomialdistribution. For nonuniform RCTs, we focus on those growing in the fashion ofpreferential attachment. We develop methods based on stochastic recurrences tocompute the exact expectations and the dispersion matrix of the degreevariables. A generalized P\'{o}lya urn model is exploited to determine theexact joint distribution of these degree variables. We apply the methods fromcombinatorics to prove that the asymptotic distribution is Dirichlet. Inaddition, we propose a new type of Gini index to quantitatively distinguish theevolutionary characteristics of the two classes of RCTs. We present the resultsvia several numerical experiments.

The degree profile and Gini index of random caterpillar trees