# Prerequisites: Some of the possible theoretical directions would benefit from B10a Martingales

## Abstract

through measure theory. If a more statistical direction is chosen, there is some (fairly elementary) programming of simulation code and code to calculate likelihoods. Part A Graph Theory may be useful but is not essential. Type: Project with possibilities for theoretical study, simulation and fitting models to data Data: Available online in database TreeBASE. Description: The alpha model was proposed by Daniel Ford as a model for phylogenetic trees that describe the genealogical tree relating species. Ford fitted the model to published phylogenies as available on the internet in a database called TreeBASE. An alpha model tree is a random rooted binary combinatorial tree Tn with n leaves (cladogram) constructed inductively as follows. For n = 1 and n = 2 there is only one such tree, looking like an I and a Y, respectively, the vertex at the bottom being the root, vertices at the top being called leaves. Given Tn, a tree Tn+1 is constructed as follows. For some parameter α ∈ (0, 1), associate weights with edges, weight 1−α each with edges between a leaf and another vertex, weight α each with all other edges. Pick an edge at random with probability mass function proportional to the weights. Add two vertices. Replace the chosen edge by two edges connecting the end points to one of the new vertices and add a further edge between the new vertices. The plan of this project is to carefully set up the model, study its properties, possibly simulate and/or fit the model to some data. There is a wide range of possibilities on the theoretical side including combinatorial and asymptotic properties, with some recent literature, to a large part at an accessible level, challenging in other parts. The data analysis side would repeat Ford’s study on some more trees and then compare with other models such as Aldous’s beta-splitting model and/or non-binary models such as the alphagamma model, where no such study has been carried out; further questions will arise while doing this