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This is a continuation of the study, begun by Ceccherini-Silberstein andWoess, of context-free pairs of groups and the related context-free graphs inthe sense of Muller and Schupp. Instead of the cones (connected components withrespect to deletion of finite balls with respect to the graph metric), a moregeneral approach to context-free graphs is proposed via tree sets consisting ofcuts of the graph, and associated structure trees. The existence of tree setswith certain "good" properties is studied. With a tree set, a naturalcontext-free grammar is associated. These investigations of the structure ofcontext free pairs, resp. graphs are then applied to study random walkasymptotics via complex analysis.

Context-free pairs of groups. II - cuts, tree sets, and random walks