On the Tree of Binary Cube-Free Words
We present two related results on the prefix tree of all binary cube-free words. First, we show that non-branching paths in this tree are short: such a path from a node of nth level has length \(O(\log n)\). Second, we prove that the lower density of the set of branching points along any infinite path is at least 23/78. Our results are based on a technical theorem describing the mutual location of “almost cubes” in a cube-free word.
KeywordsCube-free word Power-free word Prefix tree