Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s beta-splitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
"Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models." Ann. Probab. 36 (5) 1790 - 1837, September 2008. https://doi.org/10.1214/07-AOP377