  We consider conducting inference on the output of the Classification and
Regression Tree (CART) [Breiman et al., 1984] algorithm. A naive approach to
inference that does not account for the fact that the tree was estimated from
the data will not achieve standard guarantees, such as Type 1 error rate
control and nominal coverage. Thus, we propose a selective inference framework
for conducting inference on a fitted CART tree. In a nutshell, we condition on
the fact that the tree was estimated from the data. We propose a test for the
difference in the mean response between a pair of terminal nodes that controls
the selective Type 1 error rate, and a confidence interval for the mean
response within a single terminal node that attains the nominal selective
coverage. Efficient algorithms for computing the necessary conditioning sets
are provided. We apply these methods in simulation and to a dataset involving
the association between portion control interventions and caloric intake.
