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We study the formation of stationary localized states using the discretenonlinear Schr\"{o}dinger equation in a Cayley tree with connectivity $K$. Twocases, namely, a dimeric power law nonlinear impurity and a fully nonlinearsystem are considered. We introduce a transformation which reduces the Cayleytree into an one dimensional chain with a bond defect. The hopping matrixelement between the impurity sites is reduced by $1/\surd K$. The transformedsystem is also shown to yield tight binding Green's function of the Cayleytree. The dimeric ansatz is used to find the reduced Hamiltonian of the system.Stationary localized states are found from the fixed point equations of theHamiltonian of the reduced dynamical system. We discuss the existence ofdifferent kinds of localized states. We have also analyzed the formation oflocalized states in one dimensional system with a bond defect and nonlinearitywhich does not correspond to a Cayley tree. Stability of the states isdiscussed and stability diagram is presented for few cases. In all cases thetotal phase diagram for localized states have been presented.Comment: Latex file, Eight figures are available on reques

The role of power law nonlinearity in the discrete nonlinear Schrödinger equation on the formation of stationary localized states in the Cayley tree