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Preprint Number 2176
2176. Roland Walker Tree Dimension and the Sauer-Shelah Dichotomy E-mail: Submission date: 23 March 2022 Abstract: We introduce tree dimension and its leveled variant in order to measure the complexity of leaf sets in binary trees. We then provide a tight upper bound on the size of such sets using leveled tree dimension. This, in turn, implies both the famous Sauer-Shelah Lemma for VC dimension and Bhaskar's version for Littlestone dimension, giving clearer insight into why these results place the exact same upper bound on their respective shatter functions. We also classify the isomorphism types of maximal leaf sets by tree dimension. Finally, we generalize this analysis to higher-arity trees. Mathematics Subject Classification: 05C05, 05A05, 68Q32, 03C45, 03C98 Keywords and phrases: |
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