On the VC-dimension of univariate decision trees

In this paper, we give and prove lower bounds of the VC-dimension of the univariate decision tree hypothesis class. The VC-dimension of the univariate decision tree depends on the VC-dimension values of its subtrees and the number of inputs. In our previous work (Aslan et al., 2009), we proposed a search algorithm that calculates the VC-dimension of univariate decision trees exhaustively. Using the experimental results of that work, we show that our VC-dimension bounds are tight. To verify that the VC-dimension bounds are useful, we also use them to get VC-generalization bounds for complexity control using SRM in decision trees, i.e., pruning. Our simulation results shows that SRM-pruning using the VC-dimension bounds finds trees that are more accurate as those pruned using cross-validation.