  A matching complex of a simple graph $G$ is a simplicial complex with faces
given by the matchings of $G$. The topology of matching complexes is
mysterious; there are few graphs for which the homotopy type is known. Marietti
and Testa showed that matching complexes of forests are contractible or
homotopy equivalent to a wedge of spheres. We study two specific families of
trees. For caterpillar graphs, we give explicit formulas for the number of
spheres in each dimension and for perfect binary trees we find a strict
connectivity bound. We also use a tool from discrete Morse theory called the
\textit{Matching Tree Algorithm} to study the connectivity of honeycomb graphs,
partially answering a question raised by Jonsson.
