The reconstruction of mutation genealogies from present-day chromosomes can provide us with information about past events. Moving backwards in time is a fascinating goal owing to the sharp contrast between the simplicity of the question and the extreme difficulty of conducting backward statistical reconstruction of genealogies from observed haplotypes.

simple geometric structures

When modeling of the decay of haplotype sharing, simple geometric structures that can be described unambiguously in mathematical terms can provide an algebraic framework for analyses of the forces shaping the genealogy of a single allele.

few nodes trees

If we wish to obtain an image of ancestral genealogies with a useful degree of precision, we need an algebraic description of haplotype decay for any possible graph representing the ancestry of a given variant.

An algebraic description of haplotype decay for any possible graph ancestry is of visible usefulness. Haplotype decay of a given isolated segment is plainly described as a simple power function. The beginning and end of the problem is the mathematical description of the branchings.

This, in a nutshell. An algebraic solution is challenging, but possible and interesting by itself beyond mere usefulness.