Recent experimental work has described an elegant pattern of branching in the development of the lung. Multiple forms of branching have been identified, including side branching and tip bifurcation. A particularly interesting feature is the phenomenon of 'orthogonal rotation of the branching plane'. The lung must fill 3D space with the essentially 2D phenomenon of branching. It accomplishes this by rotating the branching plane by 90° with each generation. The mechanisms underlying this rotation are not understood. In general, the programmes that underlie branching have been hypothetically attributed to genetic 'subroutines' under the control of a 'global master routine' to invoke particular subroutines at the proper time and location, but the mechanisms of these routines are not known. Here, we demonstrate that fundamental mechanisms, the reaction and diffusion of biochemical morphogens, can create these patterns. We used a partial differential equation model that postulates three morphogens, which we identify with specific molecules in lung development. We found that cascades of branching events, including side branching, tip splitting and orthogonal rotation of the branching plane, all emerge immediately from the model, without further assumptions. In addition, we found that one branching mode can be easily switched to another, by increasing or decreasing the values of key parameters. This shows how a 'global master routine' could work by the alteration of a single parameter. Being able to simulate cascades of branching events is necessary to understand the critical features of branching, such as orthogonal rotation of the branching plane between successive generations, and branching mode switch during lung development. Thus, our model provides a paradigm for how genes could possibly act to produce these spatial structures. Our low-dimensional model gives a qualitative understanding of how generic physiological mechanisms can produce branching phenomena, and how the system can switch from one branching pattern to another using low-dimensional 'control knobs'. The model provides a number of testable predictions, some of which have already been observed (though not explained) in experimental work.