When lungs are emptied during exhalation, peripheral airways close up. For people with lung disease, they may not reopen for a significant portion of inhalation, impairing gas exchange. A knowledge of the mechanisms that govern reinflation of collapsed regions of lungs is therefore central to the development of ventilation strategies for combating respiratory problems. Here we report measurements of the terminal airway resistance, Rt, during the opening of isolated dog lungs. When inflated by a constant flow, Rt decreases in discrete jumps. We find that the probability distribution of the sizes of the jumps and of the time intervals between them exhibit power-law behaviour over two decades. We develop a model of the inflation process in which 'avalanches' of airway openings are seen--with power-law distributions of both the size of avalanches and the time intervals between them--which agree quantitatively with those seen experimentally, and are reminiscent of the power-law behaviour observed for self-organized critical systems. Thus power-law distributions, arising from avalanches associated with threshold phenomena propagating down a branching tree structure, appear to govern the recruitment of terminal airspaces.