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In this paper we consider two aspects of the inverse problem of how toconstruct merge trees realizing a given barcode. Much of our investigationexploits a recently discovered connection between the symmetric group andbarcodes in general position, based on the simple observation that death orderis a permutation of birth order. The first important outcome of our study is aclear combinatorial distinction between the space of phylogenetic trees (asdefined by Billera, Holmes and Vogtmann) and the space of merge trees. GenericBHV trees on $n+1$ leaf nodes fall into $(2n-1)!!$ distinct strata, but theanalogous number for merge trees is equal to the number of maximal chains inthe lattice of partitions, i.e., $(n+1)!n!2^{-n}$. The second aspect of ourstudy is the derivation of precise formulas for the distribution of treerealization numbers (the number of merge trees realizing a given barcode) whenwe assume that barcodes are sampled using a uniform distribution on thesymmetric group. We are able to characterize some of the higher moments of thisdistribution, thanks in part to a reformulation in terms of Dirichletconvolution. This characterization provides a type of null hypothesis,apparently different from the distributions observed in real neuron data andopens the door to doing more precise science.

From Trees to Barcodes and Back Again II: Combinatorial and Probabilistic Aspects of a Topological Inverse Problem