The empty word is the unique vertex of degree three in the underlying <tree>; every other vertex has degree four. Every graph automorphism therefore fixes the empty word and permutes its three neighbours. Those neighbours are precisely $\{\mathbf0,\mathbf1,\mathbf2\}$, so this set is invariant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 104 / 3 / a / Solution

 Solution (source code)