Why do Decision Trees/rpart prefer to choose continuous over categorical variables? I run some decision trees in rpart with 10 continuous variables and 3 categorical variables (with 1 or 0 options), the result of the tree was that none of the 3 categorical variables were used in building the trees. Do you know if maybe there is a deficiency combining data from continuous and categorical variables for decision trees? 
 A: Yes, classical decision tree algorithms - e.g., CART (as implemented in rpart) or C4.5 - are biased towards variables with many possible splits. The reason is that they use exhaustive search over all possible splits in all possible variables without accounting for finding larger improvements by "chance" when searching over more splits. This is addressed by various decision tree algorithms based on statistical inference, pioneered by Loh and co-workers:


*

*Loh WY, Shih YS (1997). "Split Selection Methods for Classification 
Trees." Statistica Sinica, 7, 815-840.

*Loh WY (2002). "Regression Trees with Unbiased Variable Selection and
Interaction Detection." Statistica Sinica, 12, 361-386.

*Hothorn T, Hornik K, Zeileis A (2006). "Unbiased Recursive Partitioning:
A Conditional Inference Framework." Journal of Computational and
Graphical Statistics, 15(3), 651-674.


In R the partykit package (successor to the party package) implements several unbiased decision tree algorithms. Specifically, ctree() (Hothorn et al. 2006) should be easy to try for you as an alternative for rpart.
Having said that: It is, of course, possible that even when using an unbiased decision tree algorithm none of the categorical variables appear in the tree because they are not relevant (or not relevant after adjusting for the continuous ones as suggested by @gung).
