In many plant species only a small proportion of buds yield branches. Both the timing and extent of bud activation are tightly regulated to produce specific branching architectures. For example, the primary shoot apex can inhibit the activation of lateral buds. This process is termed apical dominance and is dependent on the plant hormone auxin moving down the main stem in the polar auxin transport stream. We use a computational model and mathematical analysis to show that apical dominance can be explained in terms of an auxin transport switch established by the temporal precedence between competing auxin sources. Our model suggests a mechanistic basis for the indirect action of auxin in bud inhibition and captures the effects of diverse genetic and physiological manipulations. In particular, the model explains the surprising observation that highly branched Arabidopsis phenotypes can exhibit either high or low auxin transport.