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

Attractors of Trees of Maps and of Sequences of Maps between Spaces with Application to Subdivision