Decision trees as partitioning machines to characterize their generalization properties

Decision trees are popular machine learning models that are simple to build and easy to interpret. Even though algorithms to learn decision trees date back to almost 50 years, key properties affecting their generalization error are still weakly bounded. Hence, we revisit binary decision trees on real-valued features from the perspective of partitions of the data. We introduce the notion of partitioning function, and we relate it to the growth function and to the VC dimension. Using this new concept, we are able to find the exact VC dimension of decision stumps, which is given by the largest integer $d$ such that $2\ell \ge \binom{d}{\floor{\frac{d}{2}}}$, where $\ell$ is the number of real-valued features. We provide a recursive expression to bound the partitioning functions, resulting in a upper bound on the growth function of any decision tree structure. This allows us to show that the VC dimension of a binary tree structure with $N$ internal nodes is of order $N \log(N\ell)$. Finally, we elaborate a pruning algorithm based on these results that performs better than the CART algorithm on a number of datasets, with the advantage that no cross-validation is required.