  In this article, we construct a generalization of the Blum-Fran\c{c}ois
Beta-splitting model for evolutionary trees, which was itself inspired by
Aldous' Beta-splitting model on cladograms. The novelty of our approach allows
for asymmetric shares of diversification rates (or diversification `potential')
between two sister species in an evolutionarily interpretable manner, as well
as the addition of extinction to the model in a natural way. We describe the
incremental evolutionary construction of a tree with n leaves by splitting or
freezing extant lineages through the Generating, Organizing and Deleting
processes. We then give the probability of any (binary rooted) tree under this
model with no extinction, at several resolutions: ranked planar trees giving
asymmetric roles to the first and second offspring species of a given species
and keeping track of the order of the speciation events occurring during the
creation of the tree, unranked planar trees, ranked non-planar trees and
finally (unranked non-planar) trees. We also describe a continuous-time
equivalent of the Generating, Organizing and Deleting processes where tree
topology and branch-lengths are jointly modeled and provide code in
SageMath/python for these algorithms.
