The morphogenesis of branched tissues has been a subject of long-standing interest and debate. Although much is known about the signaling pathways that control cell fate decisions, it remains unclear how macroscopic features of branched organs, including their size, network topology and spatial patterning, are encoded. Based on large-scale reconstructions of the mouse mammary gland and kidney, we show that statistical features of the developing branched epithelium can be explained quantitatively by a local self-organizing principle based on a branching and annihilating random walk (BARW). In this model, renewing tip-localized progenitors drive a serial process of ductal elongation and stochastic tip bifurcation that terminates when active tips encounter maturing ducts. Finally, based on reconstructions of the developing mouse salivary gland, we propose a generalisation of BARW model in which tips arrested through steric interaction with proximate ducts reactivate their branching programme as constraints become alleviated through the expansion of the underlying matrix. This inflationary branching-arresting random walk model presents a general paradigm for branching morphogenesis when the ductal epithelium grows cooperatively with the matrix into which it expands.