  The problems of model and variable selections for classification trees are
jointly considered. A penalized criterion is proposed which explicitly takes
into account the number of variables, and a risk bound inequality is provided
for the tree classifier minimizing this criterion. This penalized criterion is
compared to the one used during the pruning step of the CART algorithm. It is
shown that the two criteria are similar under some specific margin assumptions.
In practice, the tuning parameter of the CART penalty has to be calibrated by
hold-out. Simulation studies are performed which confirm that the hold-out
procedure mimics the form of the proposed penalized criterion.
