Can CART models be made robust? A colleague in my office said to me today "Tree models aren't good because they get caught by extreme observations".
A search here resulted in this thread that basically supports the claim.
Which leads me to the question - under what situation can a CART model be robust, and how is that shown?
 A: You might consider using Breiman's bagging or random forests.  One good reference is Breiman "Bagging Predictors" (1996).  Also summarized in Clifton Sutton's "Classification and Regression Trees, Bagging, and Boosting" in the Handbook of Statistics.
You can also see Andy Liaw and Matthew Wiener R News discussion of the randomForest package.
A: If you check out the 'gbm' package in R (generalized gradient boosting), the 'boosting' uses loss functions that are not necessarily mean squared error.  This shows up in the 'distribution' argument to function 'gbm()'.  Thus the elaboration of the tree via boosting will be resistant to outliers, similar to how M-estimators work.
You might start here.
Another approach would be to build the tree the usual way (partitions based on SSE), but prune the tree using cross validation with a robust measure of fit.  I think xpred in rpart will give cross validated predictors (for a variety of different tree complexities), which you can then apply your own measure of error, such as mean absolute value.
A: No, not in their present forms.
The problem is that convex loss functions cannot be made to be robust to contamination by outliers (this is a well known fact since the 70's but keeps being rediscovered periodically, see for instance this paper for one recent such re-discovery):
http://www.cs.columbia.edu/~rocco/Public/mlj9.pdf
Now, in the case of regression trees, the fact that CART uses marginals (or alternatively univariate projections) can be used:
one can think of a version of CART where the s.d. criterion is replaced by a more
robust counterpart (MAD or better yet, Qn estimator).
Edit:
I recently came across an older paper implementing the approach suggested above (using robust M estimator of scale instead of the MAD). This will impart robustness to "y" outliers to CART/RF's (but not to outliers located on the design space, which will affect the estimates of the model's hyper-parameters)  See:
Galimberti, G., Pillati, M., & Soffritti, G. (2007). Robust regression trees based on M-estimators.
Statistica, LXVII, 173–190.
