This paper revisits the concept of controlled reduction, a symmetry-based method of decomposing the control of mechanical systems into lower-dimensional problems. Founded on a geometric property of serial kinematic chains termed recursive cyclicity, the present work extends this property to branched chains by defining the minimal irreducible tree structure. We then generalize the subrobot theorem to branched chains, showing that we can use control to reduce any robot's dimensionality to that of its irreducible tree structure. This method is applied to a 5-DOF bipedal robot with a hip and torso - we construct stable straight-ahead and turning gaits in 3-D space based on known sagittal-plane limit cycles.