Is there a similar categorization to Elementary Cellular Automata?
You might take a look at Lindenmayer Systems. 

From Wikipedia:


  An L-system consists of an alphabet of symbols that can be used to
  make strings, a collection of production rules that expand each symbol
  into some larger string of symbols, an initial "axiom" string from
  which to begin construction, and a mechanism for translating the
  generated strings into geometric structures.


L-systems can be used to generate a Sierpinski triangle.