A maximum parsimony tree was then generated based on these characters, and the number of steps required. A second maximum parsimony tree was generated using linguistic data from 77 Austronesian languages comprising of 5,185 morphemes. The character tree was then mapped onto the linguistically produced tree to examine fit; if the tree fit well, it suggests an Express Train model of migration. The tree generated had 13 steps, which in comparison to the maximum fit ( 9, ie n-1 ) and the number of steps expected by chance (49) showed significant support for the Express Train version of events. This paper was fairly simple, and due to the nature of the study was unable to model an effective route for the slow train model, however it displayed the effectiveness phylogenetic techniques on linguistic problems. 
Gray expanded on his evidence for the Express Train model by using phylogenetic models based on Bayesian inference, and application of MCMC algorithms. Language Phylogenies Reveal Expansion Pulses and Pauses in Pacific Settlement (2009) builds on lessons learnt in classical lexicostatistics and glottochronology: It uses a relatively short word list (210), and extensively curated it before modelling began. Linguists established cognates between 400 languages, and consulted independent linguists to confirm the accuracy of their list (90-93% accuracy, IRR=91%). 
A consensus tree was selected using a Metropolis-Hastings based algorithm, from a posterior probability distribution comprised of 4,200 trees (language evolution was modelled as single cognate gain/loss). The consensus tree provided more evidence of Dempwolff's original theory of the singular oceanic language group. The consensus tree also agreed with Grace et al's positioning of the Polynesian languages nested within the Central-Pacific subgroup. The research provides evidence for a Taiwanese origin of migration through Melanesia, and pause/pulse theory of migration, with an initial refractory period before rapid expansion out of Taiwan through Melanesia, with a second pause occurring once the Polynesians arrived at Samoa. Gray estimated the age of the Austronesian language family to be 5230 years old, nearly 8000 younger than would be predicted with the Slow Boat model, adding more evidence to the Express Train theory of Polynesian expansion. 
With the exception of these papers, little to no phylogenetic based work has been applied to Polynesian migration, however given the promising conclusions the papers have delivered, its possible that phylogenetic methods may be able to answer more and more mysteries about Polynesian expansion, migration and settlement.

Summary diagram

From all of these pieces of data, a cumulative map of Polynesian migration has been created: