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	   In this paper, we determine the homotopy types of the Morse complexes 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 its 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. (2008,  Topology and Its Applications  155, 2084–2089). 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.

On the homotopy and strong homotopy type of complexes of discrete Morse functions