  In this work we answer an open question asked by Johnson--Scoville. We show
that each merge tree is represented by a discrete Morse function on a path.
Furthermore, we present explicit constructions for two different but related
kinds of discrete Morse functions on paths that induce any given merge tree. A
refinement of the used methods allows us to define notions of equivalence of
discrete Morse functions on trees which give rise to a bijection between
equivalence classes of discrete Morse functions and isomorphism classes of
certain labeled merge trees. We also compare our results to similar ones from
the literature, in particular to work by Curry.
