Time-conditional properties of branches in coalescent gene trees

Coalescent gene trees have proven to be a powerful framework for formulating and solving problems in population genetics both in theory and practice. Using them, geneticists have been able to generate expectations for many attributes of a random sample of genotypes from a population given a model of the history of the population. This paper derives three new properties of coalescent gene trees that will help characterize the present-day impacts of historical events. Considering a single branch sampled at a given time ts in the past, it presents distributions describing 1) the length of time a branch sampled ts generations in the past had existed at the time of sampling, 2) the length of time that branch continues from time ts towards the present, and3) the the probability that the branch is ancestral to x individuals in a modern sample.