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Lately, algebraic geometry has been offering a novel approach to advancing problems on multivariate Gaussian models. This is done by identifying Gaussian distributions with symmetric matrices and analyzing the set of polynomials that vanish on these matrices, referred to as ideals. The talk will focus on Brownian motion tree (BMT) models – Gaussian models in phylogenetics. These models have a hidden toric geometry, which we use to advance questions on their maximum likelihood estimate. Lastly, motivated by the need to classify toric statistical models, we introduce the symmetry Lie group of an ideal and an associated algorithm as a means to detect non-toric structures in a general setting. No prior knowledge on toric ideals or BMT models is assumed.