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

TL;DR

This paper explores the combinatorial and probabilistic aspects of constructing merge trees from barcodes, revealing new structural distinctions and distribution formulas that enhance understanding of topological inverse problems.

Contribution

It provides a clear combinatorial distinction between phylogenetic and merge trees and derives formulas for the distribution of tree realization numbers under uniform barcode sampling.

Findings

Distinct counts of strata for BHV and merge trees.

Formulas for the distribution of realization numbers.

Higher moments characterized via Dirichlet convolution.

Abstract

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., $(n+1)!n!2^{-n}$. 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.