Multiway systems based on numbers

One can consider for example the rule n -> {n + 1, 2 n} implemented by

NestList[Union[Flatten[{# + 1, 2 #}]]&, {0}, t]

In this case there are Fibonacci[t+2] distinct numbers obtained at step t. In general, rules based on simple arithmetic operations yield only simple nested structures. If the numbers n are allowed to have both real and imaginary parts then results analogous to those discussed for substitution systems on page 933 are obtained. (Somewhat related systems based on recursive sequences are discussed on page 907. Compare also sorted multiway systems on page 937.)