  Decision trees with binary splits are popularly constructed using
Classification and Regression Trees (CART) methodology. For binary
classification and regression models, this approach recursively divides the
data into two near-homogenous daughter nodes according to a split point that
maximizes the reduction in sum of squares error (the impurity) along a
particular variable. This paper aims to study the bias and adaptive properties
of regression trees constructed with CART. In doing so, we derive an
interesting connection between the bias and the mean decrease in impurity (MDI)
measure of variable importance---a tool widely used for model
interpretability---defined as the sum of impurity reductions over all
non-terminal nodes in the tree. In particular, we show that the probability
content of a terminal subnode for a variable is small when the MDI for that
variable is large and that this relationship is exponential---confirming
theoretically that decision trees with CART have small bias and are adaptive to
signal strength and direction. Finally, we apply these individual tree bounds
to tree ensembles and show consistency of Breiman's random forests. The context
is surprisingly general and applies to a wide variety of multivariable data
generating distributions and regression functions. The main technical tool is
an exact characterization of the conditional probability content of the
daughter nodes arising from an optimal split, in terms of the partial
dependence function and reduction in impurity.
