The Mean and Variance of the Numbers of r-Pronged Nodes and r-Caterpillars in Yule-Generated Genealogical Trees

Abstract.

The Yule model is a frequently-used evolutionary model that can be utilized to generate random genealogical trees. Under this model, using a backwards counting method differing from the approach previously employed by Heard (Evolution 46: 1818–1826), for a genealogical tree of n lineages, the mean number of nodes with exactly r descendants is computed (2 ≤  r ≤  n − 1). The variance of the number of r-pronged nodes is also obtained, as are the mean and variance of the number of r-caterpillars. These results generalize computations of McKenzie and Steel for the case of r = 2 (Math. Biosci. 164: 81–92, 2000). For a given n, the two means are largest at r = 2, equaling 2n/3 for n ≥  5. However, for n ≥  9, the variances are largest at r = 3, equaling 23n/420 for n ≥  7. As n→∞, the fraction of internal nodes that are r-caterpillars for some r approaches (e² − 5)/4≈ 0.59726.