We consider the motion of a finger of low-viscosity fluid as it propagates into a branching network of fluid-filled microchannels – a scenario that arises in many appli- cations, such as microfluidics, biofluid mechanics (e.g. pulmonary airway reopening) and the flow in porous media. We perform experiments to investigate the behaviour of the finger as it reaches a single bifurcation and determine under what conditions the finger branches symmetrically. We find that if the daughter tubes have open ends, the finger branches asymmetrically and will therefore tend to reopen a single path through the branching network. Conversely, if the daughter tubes terminate in elastic chambers, which provide a lumped representation of the airway wall elasticity in the airway reopening problem, the branching is found to be symmetric for sufficiently small propagation speeds. A mathematical model is developed to explain the experimentally observed behaviour.