FIRM comes in two varieties for categorical or continuous response. Predictors can be nominal or ordinal and there is usually provision for a missing-value category. Some versions can handle continuous predictors, others cannot. There are also a variety of methods that do splits on linear combinations rather than single predictors. I have not yet constructed a taxonomy for such methods.
Some programs combine two or more families.
For example, IND combines methods from CART and C4 as well as Bayesian and minimum encoding methods. There are numerous unresolved statistical issues regarding these methods. CART supporters claim that its pruning method using cross-validation is superior to the significance testing method used in the AID family. The validity of CART cross-validation is suspect because CART seems to produce much smaller trees than the AID family, even using very conservative significance levels for the latter, which one would expect to validate well although empirical evidence is scarce. I have not seen any published comparison of CART and AID methods.