  Identifiability is a crucial property for a statistical model since
distributions in the model uniquely determine the parameters that produce them.
In phylogenetics, the identifiability of the tree parameter is of particular
interest since it means that phylogenetic models can be used to infer
evolutionary histories from data. In this paper we introduce a new
computational strategy for proving the identifiability of discrete parameters
in algebraic statistical models that uses algebraic matroids naturally
associated to the models. We then use this algorithm to prove that the tree
parameters are generically identifiable for 2-tree CFN and K3P mixtures. We
also show that the $k$-cycle phylogenetic network parameter is identifiable
under the K2P and K3P models.
