The study of periodic Jacobi matrices on the line is a classical subject in spectral theory, motivated by condensed matter physics and having connections to orthogonal polynomial theory, potential theory and other areas of mathematics. In this talk we will discuss ongoing work that attempts to generalize the theory to more general trees, with an emphasis on Floquet theory which describes the structure of generalized eigenfunctions. We will describe some results obtained in joint works with Jess Banks, Jorge Garza Vargas, Eyal Seelig and Barry Simon.