Branched elastic rod structures are abundant in engineering and nature, in applications ranging from MEMS devices to human spine models. While buckling is well-understood for problems of this type, stability is often difficult to assess, especially when the model is derived from a nonlinear rod theory. The purpose of this research is to establish criteria for determining nonlinear stability, based upon the minimization of an energy functional. By utilizing variational principles, and Legendre's classical work in particular, a new necessary condition for stability featuring the existence of bounded solutions to a set of Riccati differential equations is established. For a single rod, building on classical results, this condition is also shown to be sufficient for stability. The stability criteria are demonstrated on a number of examples using a simple, planar rod theory. These examples range from a classical strut under axial load to a branched tree-like structure composed of several rods. In the branched model, the stability analysis consists of finding bounded solutions to a set of Riccati equations, which are coupled at branching points. The number of Riccati equations corresponds to the number of rods in the structure. The resulting condition is only necessary for stability of a branched structure, as a sufficient condition could not be established. However, this is the first instance of a stability criterion for branched structures that is based on the second variation of the total energy. The advantage is that this method provides a systematic means of identifying unstable, and therefore physically unrealizable, configurations of a branched structure. Finally, an extension of the stability criteria to other rod theories is discussed.