Running a logistic regression we get p-values for all the input variables which helps us choose significant inputs. Similarly can we use the classification trees to pick variables that are split, and use those variables in the model? I think the fact that splitting the dataset on a variable leads to lower classification error should be a good indicator of the predictive power of the variable, is that true?
It is generally not valid to "choose significant inputs", a form of highly problematic stepwise regression. Using classification trees to form inputs for logistic regression is even more problematic because of simultaneously increasing both type I and type II error.