Following Lyons (1990, Random Walks and Percolation on Trees) we define a periodic tree, restate its branching number and consider a biased random walk on it. In the case of a transient walk, we describe the walk-invariant random periodic tree and calculate the asymptotic rate of escape (speed) of the walk. This is achieved by exploiting the connections between random walks and electric networks.
"Random Walk on Periodic Trees." Electron. J. Probab. 2 1 - 16, 1997. https://doi.org/10.1214/EJP.v2-15