Probabilities on cladograms: introduction to the alpha model

The alpha model, a parametrized family of probabilities on cladograms (rooted binary leaf labeled trees), is introduced. This model is Markovian self-similar, deletion-stable (sampling consistent), and passes through the Yule, Uniform and Comb models. An explicit formula is given to calculate the probability of any cladogram or tree shape under the alpha model. Sackin's and Colless' index are shown to be $O(n^{1+\alpha})$ with asymptotic covariance equal to 1. Thus the expected depth of a random leaf with $n$ leaves is $O(n^\alpha)$. The number of cherries on a random alpha tree is shown to be asymptotically normal with known mean and variance. Finally the shape of published phylogenies is examined, using trees from Treebase.