  In this paper, we study merge trees induced by a discrete Morse function on a
tree. Given a discrete Morse function, we provide a method to constructing an
induced merge tree and define a new notion of equivalence of discrete Morse
functions based on the induced merge tree. We then relate the matching number
of a tree to a certain invariant of the induced merge tree. Finally, we count
the number of merge trees that can be induced on a star graph and characterize
the induced merge tree.
