Advisor: Vladimir Minin Branching processes are a class of continuous-time Markov chains (CTMCs) frequently used in stochastic modeling with ubiquitous applications. One-dimensional cases such as birth-death processes are well studied, but it is often necessary to model systems with more than one species --- bivariate or other multi-type processes are commonly used to model phenomena such as competition, predation, or infection. These models often feature large or uncountable state spaces, rendering general CTMC techniques to compute transition probabilities such as matrix exponentiation and simulation-based approaches impractical. We present spectral techniques to compute transition probabilities comprising the observed data likelihood for discretely and unevenly observed multi-type branching processes, enabling likelihood-based inference. Our technique reduces these calculations to low dimensional integration, and analogously enables calculation of related terms such as expected sufficient statistics within an expectation maximization (EM) algorithm. We assess robustness, accuracy and efficiency of our EM algorithm in several simulation studies applied to a birth-death-shift (BDS) model, and apply it to estimate intrapatient time evolution of IS6110 transposable element, a genetic marker frequently used during epidemiological studies of Mycobacterium tuberculosis. Finally, we incorporate our methods for computing transition probabilities within a compressed sensing framework, demonstrating scalability in the presence of sparsity.