Mismatch between significant variables from logistic regression and tree splits in R I'm studying a data set in R using both regression trees (tree and rpart functions) and logistic regression. I'm finding explanatory variables in the regression which are significant, but when I fit a tree those variables are not used as splits. What does this imply about the predictive ability of the results?
 A: Simply said: there's more than one way to skin a cat. Logistic regression and regression trees are simply different algorithms that may pick up different variables as more or less important, because they are used in a different way.
Simply said, in logistic regression, the influence of all variables is studied as they work simultaneously, while trees look at the influence of the variables one at a time (more or less).
If a combination of, say, 2 variables is very indicative of the result, but each individual variable is not, this will typically not be picked up in regression tree type settings. The easiest example of this is:
out  pred1  pred2
1    1      1
1    0      0
0    1      0
0    0      1

You can easily see that both pred1 and pred2 are completely uninformative for out, but the combination of the two is completely informative (out is 1 if both variables are the same). I doubt this will get picked up in regression trees, but logistic regression (well, a penalized version of it, but that's another story) would.
I'm confident that examples can be constructed where regression trees would have the benefit (probably where two variables are highly correlated). 
So in short: this probably mainly depends on the correlation structure of your predictors. The rule of thumb for these matters is: if you're looking for association (statistical significance), avoid correlation between your predictors. If you're looking for a model with high predictive value: never mind (but be aware that you may be overfitting).
A: It does not say anything about the predictive ability of the model. It just says that the response varies differently as you advance in different direction in the predictor space. In means in some directions, the response seems to be non-continuous (thus the split), and in another direction it seems to have an S shaped smooth variation (the logistic function). The logistic and tree fits, might achieve the same goodness of fit. 
Going a step further- you might want to consider staying in the logistic regression framework but dealing with the non-continuous predictor by:
- a semi parametric approach allowing a transformation of the split variable.
- creating an indicator for the different regions of the split variable and allowing it to interact with the original variable. 
A: Both of the methods you've used are highly unstable and not trustworthy.  So I wouldn't worry about their disagreements so much.  And single trees are seldom accurate.  Their huge instability is why bagging, boosting, and random forests were created.
