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We study colorings of regular trees using subball complexity b(n), which is the number of colored n-balls up to color-preserving isomorphisms. We show that for any k-regular tree, for k > 1, there are colorings of intermediate complexity. We then construct colorings of linear complexity b(n) = 2n + 2. We also construct colorings induced from sequences of linear subword complexity which has exponential subball complexity.

COLORINGS OF TREES WITH LINEAR, INTERMEDIATE AND EXPONENTIAL SUBBALL COMPLEXITY