  The aim of this review is to present and analyze the probabilistic models of
mathematical phylogenetics which have been intensively used in recent years in
biology as the cornerstone of attempts to infer and reconstruct the ancestral
relationships between species. We outline the development of theoretical
phylogenetics, from the earliest studies based on morphological characters,
through to the use of molecular data in a wide variety of forms. We bring the
lens of mathematical physics to bear on the formulation of theoretical models,
focussing on the applicability of many methods from the toolkit of that
tradition -- techniques of groups and representations to guide model
specification and to exploit the multilinear setting of the models in the
presence of underlying symmetries; extensions to coalgebraic properties of the
generators associated to rate matrices underlying the models, in relation to
the graphical structures (trees and networks) which form the search space for
inferring evolutionary trees. Aspects presented, include relating model classes
to relevant matrix Lie algebras, as well as manipulations with group characters
to enumerate various natural polynomial invariants, for identifying robust,
low-parameter quantities for use in inference. Above all, we wish to emphasize
the many features of multipartite entanglement which are shared between
descriptions of quantum states on the physics side, and the multi-way tensor
probability arrays arising in phylogenetics. In some instances, well-known
objects such as the Cayley hyperdeterminant (the `tangle') can be directly
imported into the formalism -- for models with binary character traits, and
triplets of taxa. In other cases new objects appear, such as the remarkable
quintic `squangle' invariants for quartet tree discrimination and DNA data,
with their own unique interpretation in the phylogenetic modeling context.
