A Bratteli diagram is a type of graph in which the vertices are split into finite subsets occupying an infinite sequence of levels, starting with a bottom level and moving to successively higher levels along edges connecting consecutive levels. An information source on a Bratteli diagram consists of a sequence of PMFs on the vertex sets at each level that are compatible under edge transport. By imposing a regularity condition on the Bratteli diagram, we obtain various results for its information sources including ergodic and entropy rate decomposition theorems, a Shannon-Mcmillan-Breiman theorem, and lossless and lossy source coding theorems. Proof methodology exploits the Vershik transformation on the path space of a Bratteli diagram. Some results for finite alphabet stationary sequential information sources are seen to be a special case of the results of this paper.