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

Random boosting and random^2 forests -- A random tree depth injection approach