The fiber of persistent homology for trees

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