  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 $2\ell \ge \binom{d}{\left\lfloor\frac{d}{2}\right\rfloor}$, 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.
