We consider disordered tight-binding models which Green's functions obey the self-consistent  cavity equation. Based on these equations and the replica trick with the one-step replica symmetry breaking, we derive an analytical expression for the  fractal dimension D_{1} that distinguishes between the extended ergodic (D_{1}=1), and extended non-ergodic (multifractal) (0<D_{1}<1) states and prove the existence of the latter in a broad range of parameters as well as existence of transitions between the extended ergodic and extended non-ergodic states. The results are applied to the systems with local tree structure (random regular graphs) and to the systems with infinite connectivity (Rosenzweig-Poter random matrix theory). They are potentially important for the problem of many-body localization, as the Fock space of interacting systems has a local tree structure. The non-ergodic extended phase in the many-body systems would imply a new state of matter in which usual Boltzmann statistics does not apply as in MBL phases and which in many respects is similar to the glass.