  Subtree number index $\emph{STN}(G)$ of a graph $G$ is the number of nonempty
subtrees of $G$. It is a structural and counting based topological index that
has received more and more attention in recent years. In this paper we first
obtain exact formulas for the expected values of subtree number index of random
polyphenylene and spiro chains, which are molecular graphs of a class of
unbranched multispiro molecules and polycyclic aromatic hydrocarbons. Moreover,
we establish a relation between the expected values of the subtree number
indices of a random polyphenylene and its corresponding hexagonal squeeze. We
also present the average values for subtree number indices with respect to the
set of all polyphenylene and spiro chains with $n$ hexagons.
