From trees to barcodes and back again II: Combinatorial and probabilistic aspects of a topological inverse problem

In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. The first important outcome of our study is a clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees. Generic BHV trees on n+ 1 leaf nodes fall into (2n− 1)!! distinct strata, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data and opens the door to doing more precise science.