We study spaces of continuous functions on limits of inverse systems of compact spaces, where the bonding mappings are fully closed. A mapping between Hausdorff compacta is called fully closed if the intersection of the images of any two closed disjoint subsets is finite. We give a characterization of such systems in terms of a relation between the space of continuous functions on the limit and continuous functions on certain types of trees. When, moreover, the fibers of neighboring bonding mappings are metrizable, such systems are known as Fedorchuk compact spaces. The stated property allows us to obtain locally uniformly rotund renormings on the spaces of continuous functions on a certain subclass of Fedorchuk compacta.