  We introduce a new notion of equivalence of discrete Morse functions on
graphs called persistence equivalence. Two functions are considered persistence
equivalent if and only if they induce the same persistence diagram. We compare
this notion of equivalence to other notions of equivalent discrete Morse
functions. We then compute an upper bound for the number of persistence
equivalent discrete Morse functions on a fixed graph and show that this upper
bound is sharp in the case where our graph is a tree. This is a version of the
"realization problem" of the persistence map. We conclude with an example
illustrating our construction.
