  In this paper, we investigate the degree profile and Gini index of random
caterpillar trees (RCTs). We consider RCTs which evolve in two different
manners: uniform and nonuniform. The degrees of the vertices on the central
path (i.e., the degree profile) of a uniform RCT follow a multinomial
distribution. For nonuniform RCTs, we focus on those growing in the fashion of
preferential attachment. We develop methods based on stochastic recurrences to
compute the exact expectations and the dispersion matrix of the degree
variables. A generalized P\'{o}lya urn model is exploited to determine the
exact joint distribution of these degree variables. We apply the methods from
combinatorics to prove that the asymptotic distribution is Dirichlet. In
addition, we propose a new type of Gini index to quantitatively distinguish the
evolutionary characteristics of the two classes of RCTs. We present the results
via several numerical experiments.
