How is AUC of decision tree calculated? I have a dataset which only has one continuous variable, and I try to use decision tree algorithm to build a model which classify the +ve and -ve label from the dataset. I run 10-fold cross-validation.
How is the AUC calculated for the decision tree classifier? Will the algorithm check different threshold value of the classifier, and determine the AUC?
What about I have more than 2 continuous variable?
 A: Suppose you predict with your model a test set $X_T = \{x_1, x_2, .., x_t\}$. Usually the prediction result for classification is a list of labels $C_T = \{c_1, c_2, .., c_t\}$, where $c_i = {+ve}|{-ve}$.
If instead of $C_T$ you obtain the distribution of these labels $D_T = \{d_1, ..,d_t\}$ where $d_i = (P(c_i = +ve), P(c_i = -ve))$ then you can compute AUC or ROC curves. Basically $P$ does not need to be probability, it is enough to be a score of some sort.
In the case of a simple DT a score could be computed from the proportion of target cases in each node. In case of a RandomForests the score could be the proportion of predictions given by each tree.
In short it does not matter what kind of binary classifier it is. The only requirement for that classifier is to be able to compute a score, instead of plain label prediction.
Now, if you have scores for each test observation, the way you build ROC curves and/or AUC is to slide a threshold from the minimum score to maximum score, in a way that all scores which pass the threshold value are classified with one label and the others with the remaining label.
The most comprehensive and crystal clear paper on this topic I found to be the one authored by Fawcett ROC Graphs: Notes and Practical Considerations for Researchers. You have to give it some time but it will be a well invested time.  
