  We introduce a novel interpretable and tree-based algorithm for prediction in
a regression setting in which each tree in a classical random forest is
replaced by a family of planted trees that grow simultaneously. The motivation
for our algorithm is to estimate the unknown regression function from a
functional ANOVA decomposition perspective, where each tree corresponds to a
function within that decomposition. Therefore, planted trees are limited in the
number of interaction terms. The maximal order of approximation in the ANOVA
decomposition can be specified or left unlimited. If a first order
approximation is chosen, the result is an additive model. In the other extreme
case, if the order of approximation is not limited, the resulting model puts no
restrictions on the form of the regression function. In a simulation study we
find encouraging prediction and visualisation properties of our random planted
forest method. We also develop theory for an idealised version of random
planted forests in the case of an underlying additive model. We show that in
the additive case, the idealised version achieves up to a logarithmic factor
asymptotically optimal one-dimensional convergence rates of order $n^{-2/5}$.
