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. As haplotype decay is essentially stochastic, simple geometric structures that can be described unambiguously in mathematical terms can provide the algebraic framework for untangling the forces shaping the genealogy of a single allele.
In a mutation history tree, branch length represents time or successive meioses. The power function, (1-t)n is a simple function, that yields interesting figures when the need to normalize the algebraic expressions is taken into consideration as illustrated above.
Branching points are more likely to occur when the population is expanding, and are surrogate markers for such periods. Calculating a statistical picture of the ancestral shapes of genetic trees is possible, but computer intensive.
An algebraic descriptions of the stochastic pattern around identical-by-descent sequences is required as geometric tools for analysis ancestry needed for medical genetics (recent mutations are more likely to be disease-causing) and human molecular history.