Why many decision trees are using Chi2 or Information Gain Ratio to split the node when they can directly use accuracy, lift or AUC?
Many users of decision trees are comfortable and familiar with using and interpreting chi-squared tests and hence find it easy to use them in connection with decision trees as well. Of course, there are other ways to capture associations between the response and the input variables, but usually no single one measure outperforms all (or at least most) others on all (or at least most) datasets. So many researchers tend to prefer the measure that they are comfortable with or that performs particularly well on their dataset.
Another reason for using chi-squared tests is that they provide a means to separate the splitting variable selection from the split point selection. The work of Loh and co-workers (e.g., on QUEST and GUIDE) has shown that this can avoid so-called variable selection bias. This bias means that many exhaustive search strategies prefer splitting variables with many possible splits. Hence, trees based on statistical inference is used in many statistical decision tree algorithms (e.g., CTree).