In a Random Forest model, are all possible variable combinations accounted for? To preface this question, I am building a Random Forest model using the randomForest package in R. I am not sure if this question is dependent upon my program, or if it can be answered based on the inherent properties of the algorithm. I felt this question was more appropriate for "Cross Validated" versus "Overflow", please let me know if you think otherwise.
A Random Forest model creates many decision trees which contain a subset of variables and data. Is there any guarantee that every possible combination of variables are accounted for across all of the decision trees?
For example, I have 4 variables that I am feeding into the RF model. Three of those variables are binary (X1, X2, Y1, Y2, Z1, Z2). The other variable (of ordinal type) contains 8 unique values.(A1...A8). 
This leaves me with 56 possible variable combinations.
How can I guarantee that all possible unique combinations of variables are accounted for across trees? I understand that simply increasing the number of trees would increase the likelihood of this. I also realize that my example contains a relatively small number of variables, however consider a situation with many variables and unique combinations.
 A: No, there's no such garanty. The total number of combinations gets very big, very fast.
The additional questions is: is that even necessary? Given that real data has noise and can present high correlation, forcing all combinations doesn't seem to help much.
If you know (by field experience) that all your variables are important, then maybe a deep neural net could work better.
A: 
A Random Forest model creates many decision trees which contain a
  subset of variables and data. Is there any guarantee that every
  possible combination of variables are accounted for across all of the
  decision trees?

No. Consider the extreme example where your forest consists of a single tree with depth one.

How can I guarantee that all possible unique combinations of variables
  are accounted for across trees? I understand that simply increasing
  the number of trees would increase the likelihood of this. I also
  realize that my example contains a relatively small number of
  variables, however consider a situation with many variables and unique
  combinations.

You can't. And, perhaps more importantly, you probably don't want that as well. Doing that would decrease the variance of the models.

Having said that, you could try bagged trees (without attribute bagging, i.e. random selection of features) instead of Random Forests ensuring that all combinations are visited at least once, but I'm not sure this model would perform any better than a full Random Forest.
