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?
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):
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).
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.