  In this work, we give a combinatorial-geometric model for moduli spaces of
discrete Morse functions on CW complexes. We investigate properties of moduli
spaces of discrete Morse functions and their relationship to the spaces of
smooth Morse functions on manifolds, as well as spaces of discrete Morse
matchings on CW complexes. Moreover, we provide a similar model for the moduli
space of merge trees and realise the induced merge tree and the induced barcode
as continuous maps between moduli spaces.
