We contribute to the study of the set of ternary square-free words. Under the prefix order, this set forms a tree, and we prove two results on its structure. 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 any infinite path in the tree has a lot of branching points: the lower density of the set of such points is at least 2/9.
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