The Julia set of the map $z \mapsto z^2+i$ is a dendrite fractal. I would like to know which affine maps (other than identity) map this region to a subset of itself. I imagine there are two three generators, but maybe there are more. Perhaps I am after an "iterated function system" which will generate the dendrite fractal (possibly as the limit of trees).



What you are after is really the combinatorics of the branched cover $z\mapsto z^2+i$ on the Julia set $J$. Namely, you really want to consider $J$ as a dynamical system, and not just as a topological space. As a mere topological space, the set of homeomorphisms of $J$ is HUGE.
This is a little bit similar to the characterization of the Cantor set as the unique compact metrizable zerodimensional space with no isolated points. 


OK, as hinted in my comment. Here is the fractal $J$: Now choose a branch of the squareroot so that $\sqrt{wi}$ is continuous on this set. Here is the image of $J$ under the map $\sqrt{wi}$ in green, and the image of $J$ under the map $\sqrt{wi}$ in red:
Thus $J$ is the attractor of a certain IFS (but not using affine maps). 

