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We revisit binary decision trees from the perspective of partitions of thedata. We introduce the notion of partitioning function, and we relate it to thegrowth function and to the VC dimension. We consider three types of features:real-valued, categorical ordinal and categorical nominal, with different splitrules for each. For each feature type, we upper bound the partitioning functionof the class of decision stumps before extending the bounds to the class ofgeneral decision tree (of any fixed structure) using a recursive approach.Using these new results, we are able to find the exact VC dimension of decisionstumps on examples of $\ell$ real-valued features, which is given by thelargest integer $d$ such that $2\ell \ge \binom{d}{\lfloor\frac{d}{2}\rfloor}$.Furthermore, we show that the VC dimension of a binary tree structure with$L_T$ leaves on examples of $\ell$ real-valued features is in $O(L_T\log(L_T\ell))$. Finally, we elaborate a pruning algorithm based on theseresults that performs better than the cost-complexity and reduced-error pruningalgorithms on a number of data sets, with the advantage that nocross-validation is required.

Generalization Properties of Decision Trees on Real-valued and Categorical Features