  The induction of additional randomness in parallel and sequential ensemble
methods has proven to be worthwhile in many aspects. In this manuscript, we
propose and examine a novel random tree depth injection approach suitable for
sequential and parallel tree-based approaches including Boosting and Random
Forests. The resulting methods are called \emph{Random Boost} and
\emph{Random$^2$ Forest}. Both approaches serve as valuable extensions to the
existing literature on the gradient boosting framework and random forests. A
Monte Carlo simulation, in which tree-shaped data sets with different numbers
of final partitions are built, suggests that there are several scenarios where
\emph{Random Boost} and \emph{Random$^2$ Forest} can improve the prediction
performance of conventional hierarchical boosting and random forest approaches.
The new algorithms appear to be especially successful in cases where there are
merely a few high-order interactions in the generated data. In addition, our
simulations suggest that our random tree depth injection approach can improve
computation time by up to 40%, while at the same time the performance losses in
terms of prediction accuracy turn out to be minor or even negligible in most
cases.
