In 1961 Robert Vaught published a ground breaking paper on countable model theory entitled “Denumerable Models of Complete Theories”. In this paper he set out many of the fundamental results concerning countable model theory. At the end of this paper he made a conjecture that every countable first order theory in every model of set theory had either countably or continuum many countable models. This conjecture is what is now called “Vaught’s Conjecture” and is one of the oldest open problems in model theory. One of the most significant partial results regarding Vaught's conjecture was made by Michael Morley in the 1970 paper "The Number of Countable Models". In this paper he proved that every sentence of L_{\omega_1,\omega} has either countably many, continuum many or \omega_1 many countable models. In the proof he showed that the countable models of such a sentence could be arranged in a natural way in what is now called a Vaught tree. He further showed that the number of models at any level of this tree is either countable or has size of the continuum. Hence, the only way that that Vaught's conjecture could fail is if there is a sentence of L_{\omega_1,\omega} whose Vaught tree has only countably many models at each level and has height \omega_1. In this series of talks we will present a method which (with a few assumptions) allows us to construct for each ordinal \alpha a sentence of L_{\omega_1,\omega} whose Vaught tree has height approximately \alpha and which has only countably many models at each level. As such these sentences can be viewed as approximations to a counterexample to Vaught's conjecture. This work is based on the main result of my PhD thesis.