The fiber of persistent homology for trees

Consider the space of continuous functions on a geometric tree X whose persistent homology gives rise to a finite generic barcode D . We show that there are exactly as many path connected components in this space as there are merge trees whose barcode is D . We find that each component is homotopy equivalent to a configuration space on X with specialised constraints encoded by the merge tree. For barcodes D with either one or two intervals, our method also allows us to compute the homotopy type of this space of functions.