The strand space model is one of the most successful and widely used formalisms for analysing security protocols. This might seem surprising given that the model is not able to reflect choice points in a protocol execution: the key concept in the strand space model is that of a bundle, which models exactly one possible execution of a security protocol. Inspired by the branching processes of Petri nets, we show that branching can be introduced into the strand space model in a very natural way: bundles can be generalized to branching bundles, which are able to capture several conflicting protocol executions. Our investigations of the theory of branching bundles will motivate the concept of symbolic branching bundles, and culminate in the result that every protocol has a strand space semantics in terms of a largest symbolic branching bundle. We hope our results provide a strong theoretical basis for comparing models and providing process calculi semantics in security protocol analysis. Altogether our work is related but different to a series of works by Crazzolara and Winskel. Throughout we will profit from a close relationship of the strand space model to event structures, which has already been pointed out by these authors.