  In a previous paper we considered a sequence of maps on a complete metric
space $(X,d)$ and derived an extension of the Banach fixed point theorem. We
showed that backward trajectories of maps $X\to X$ converge under mild
conditions and that they can generate new types of attractors such as scale
dependent fractals. Here we present two generalisations of this result and some
potential applications. First, we study the structure of an infinite tree of
maps $X\to X$ and discuss convergence to a unique "attractor" of the tree. We
also consider "staircase" sequences of maps, that is, we consider a countable
sequence of metric spaces $\{(X_i,d_i)\}$ and an associated countable sequence
of maps $\{T_i\}$, $T_i:X_{i}\to X_{i-1}$. We examine conditions for the
convergence of backward trajectories of the $\{T_i\}$ to a unique attractor. An
example of such trees of maps are trees of function systems leading to the
construction of fractals which are both scale dependent and location dependent.
The staircase structure facilitates linking all types of linear subdivision
schemes to attractors of function systems.
