In this talk I plan to give a brief exposition of two combinatorial objects that are of interest to computational biologists. No biology knowledge is expected; the majority of the talk will be pure math.

Part 1: Tropical Grassmannian and phylogenetic trees.
The positive tropical grassmannian Trop+Gr(2,n) can be stratified into cells labeled by trivalent trees, which produces interesting combinatorics on its own, but also turns out to be useful in epidemiologists' computations.

Part 2: Boolean networks and polynomial ideals.
One of the most common applications of computational algebra is solving polynomial equations via Grobner basis computation. Certain systems can be modeled by collections of functions over finite fields, for which solving polynomial equations gives insight into ways to control the dynamics of the system.