  In this paper, we determine the homotopy type of the Morse complex of certain
collections of simplicial complexes by studying dominating vertices or strong
collapses. We show that if $K$ contains two leaves that share a common vertex,
then the Morse complex is strongly collapsible and hence has the homotopy type
of a point. We also show that the pure Morse complex of a tree is strongly
collapsible, thereby recovering as a corollary a result of Ayala et al. In
addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is
the Morse complex of the join $K*L$. This result is used to compute the
homotopy type of the Morse complex of some families of graphs, including
Caterpillar graphs, as well as the automorphism group of a disjoint union for a
large collection of disjoint complexes.
