Fractals have no inherent dependency on microprocessors. Microprocessors can be used to generate fractals simply because they can execute the generation rules of a fractal, but you could could likewise generate them with anything which can perform a specific affine transformation, for instance a copy machine. An affine transformation is simply a linear transformation together with a translation, for fractals the linear transformation would typically be a scaling composed with a rotation. You could easily imagine how you might follow such a simple growth pattern, just iterate the same scale rotate and translate operation mechanically so the branches look like the whole thing.

I wasn't really suggesting that microprocessors are required to generate fractals, but rather that I don't understand how the process that eventually generates the fractal shapes occurs in plants. You have a good point that all that is needed is a relatively simple operation repeated in every part of the system which indeed makes it easier for me to imagine how the shape forms. From what level this operation starts to be expressed is probably something botanists can answer.

Warning I probably don't know what the hell I'm talking about. However my gut instinct is that self similarity mean shorter code (dna) shorter dna (protein chains) take less energy to copy thus gives you a survival advantage (often survival for plants means getting set up and leaves out fast). An experiment was done a while back where they took a bacterium and 'unrolled the code' of the dna (seems there are deduplication routines in nature). When the result was grown in a lab it behaved just like the original but didnt grow as large.