Summary: We study the structure of radially symmetric standing waves for the nonlinear Schrödinger equation with harmonic potential, which arises in a wide variety of applications and is known as the Gross-Pitaevskii equation in the context of Bose-Einstein condensates with parabolic traps. Both global and local bifurcation behaviour are determined showing the existence of infinitely symmetric localized states. In particular, our theory provides a theoretical proof of the existence of a solution with prescribed numbers of zeros depending on the frequency of the wave. After a few remarks concerning the critical case, numerical computations are finally presented in order to provide an illustration of the theoretical results that have been obtained and also to investigate the supercritical case for which only few results are known.