The main focus of this thesis is the study of a special class of bicritical rational maps of the Riemann sphere. This special property will be called clustering; which informally is when a subcollection of the immediate basins of the two (super-)attracting periodic orbits meet at a periodic point p, and so the basins of the attracting periodic orbits are clustered around the points on the orbit of p. Restricting ourselves to the cases where p is fixed or of period 2, we investigate the structure of such maps combinatorially; in particular showing a very simple collection of combinatorial data is enough to define a rational map uniquely in the sense of Thurston. We also use the language of symbolic dynamics to investigate pairs (f, g) of polynomials such that f - g has a fixed or period two cluster point. We find that that the internal addresses of such maps follow very definite patterns which can be shown to hold in general.