I will discuss geometric methods of investigating phylogenetic trees. In a joint project with Weronika Buczynska we investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. They have Gorenstein terminal singularities and are Fano varieties. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space whose coordinates are indexed by subsets of their leaves.