  Given a graph $G$, we study the number of independent sets in $G$, denoted
$i(G)$. This parameter is known as both the Merrifield-Simmons index of a graph
as well as the Fibonacci number of a graph. In this paper, we give general
bounds for $i(G)$ when $G$ is bipartite and we give its exact value when $G$ is
a balanced caterpillar. We improve upon a known upper bound for $i(T)$ when $T$
is a tree, and study a conjecture that all but finitely many positive integers
represent $i(T)$ for some tree $T$. We also give exact values for $i(G)$ when
$G$ is a particular type of benzenoid.
