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

Risk Bounds for Embedded Variable Selection in Classification Trees