In this thesis we study two unary stochastic abelian networks: random walk with local memory, and branching processes in a Markovian environment. The first part is joint work with Swee Hong Chan, Lionel Levine, and Boyao Li. We prove that under certain conditions the environment of the walker forms a Markov chain, and we present a stationary distribution called the oriented wired spanning forest plus one edge. We also prove a scaling limit to Brownian motion for certain walks. In the second part, we generalize branching processes through the introduction of a Markovian environment. We prove that, as with ordinary branching processes, the long-term extinction behavior is governed by a single quantity: the long-term average number of offspring produced when a single individual reproduces. We also prove a weak law of large numbers for the population as a function of time.