This dissertation studies the model theory of asymptotic classes of finite trees. By asymptotic classes, we mean classes of finite structures in a given first-order language, where we can control the sizes of definable sets. We construct asymptotic classes of finite trees. Also, we study the MS-measurability of expansions of trees. An MS-measurable structure, defined by Macpherson and Steinhorn in [16], is an infinite structure whose definable sets are equipped with an N-valued dimension and an R-valued measure satisfying natural properties. We introduce a strong notion of coordinatization, and we prove the main theorem saying that one can lift MS-measurability to coordinatized structures. This gives new examples of MS-measurable structures.