  Methods of topological data analysis have been successfully applied in a wide
range of fields to provide useful summaries of the structure of complex data
sets in terms of topological descriptors, such as persistence diagrams. While
there are many powerful techniques for computing topological descriptors, the
inverse problem, i.e., recovering the input data from topological descriptors,
has proved to be challenging. In this article we study in detail the
Topological Morphology Descriptor (TMD), which assigns a persistence diagram to
any tree embedded in Euclidean space, and a sort of stochastic inverse to the
TMD, the Topological Neuron Synthesis (TNS) algorithm, gaining both theoretical
and computational insights into the relation between the two. We propose a new
approach to classify barcodes using symmetric groups, which provides a concrete
language to formulate our results. We investigate to what extent the TNS
recovers a geometric tree from its TMD and describe the effect of different
types of noise on the process of tree generation from persistence diagrams. We
prove moreover that the TNS algorithm is stable with respect to specific types
of noise.
