  The Discrete Nonlinear Schr$\ddot{o}$dinger Equation is used to study the
formation of stationary localized states due to a single nonlinear impurity in
a Caley tree and a dimeric nonlinear impurity in the one dimensional system.
The rotational nonlinear impurity and the impurity of the form $-\chi \mid C
\mid^{\sigma}$ where $\sigma$ is arbitrary and $\chi$ is the nonlinearity
parameter are considered. Furthermore, $\mid C \mid$ represents the absolute
value of the amplitude. Altogether four cases are studies. The usual Greens
function approach and the ansatz approach are coherently blended to obtain
phase diagrams showing regions of different number of states in the parameter
space. Equations of critical lines separating various regions in phase diagrams
are derived analytically. For the dimeric problem with the impurity $-\chi \mid
C \mid^{\sigma}$, three values of $\mid \chi_{cr} \mid$, namely, $\mid
\chi_{cr} \mid = 2$, at $\sigma = 0$ and $\mid \chi_{cr} \mid = 1$ and
$\frac{8}{3}$ for $\sigma = 2$ are obtained. Last two values are lower than the
existing values. Energy of the states as a function of parameters is also
obtained. A model derivation for the impurities is presented. The implication
of our results in relation to disordered systems comprising of nonlinear
impurities and perfect sites is discussed.
