  In chemical graph theory, caterpillar trees have been an appealing model to
represent the molecular structures of benzenoid hydrocarbon. Meanwhile,
topological index has been thought of as a powerful tool for modeling
quantitative structure-property relationship and quantitative
structure-activity between molecules in chemical compounds. In this article, we
consider a class of caterpillar trees that are incorporated with randomness,
called random caterpillars, and investigate several popular topological indices
of this random class, including Zagreb index, Randi\'{c} index and Wiener
index, etc. Especially, a central limit theorem is developed for the asymptotic
distribution of the Zagreb index of random caterpillars.
