  We propose some extensions to semi-parametric models based on Bayesian
additive regression trees (BART). In the semi-parametric BART paradigm, the
response variable is approximated by a linear predictor and a BART model, where
the linear component is responsible for estimating the main effects and BART
accounts for non-specified interactions and non-linearities. Previous
semi-parametric models based on BART have assumed that the set of covariates in
the linear predictor and the BART model are mutually exclusive in an attempt to
avoid poor coverage properties and reduce bias in the estimates of the
parameters in the linear predictor. The main novelty in our approach lies in
the way we change the tree-generation moves in BART to deal with this bias and
resolve non-identifiability issues between the parametric and non-parametric
components, even when they have covariates in common. This allows us to model
complex interactions involving the covariates of primary interest, both among
themselves and with those in the BART component. Our novel method is developed
with a view to analysing data from an international education assessment, where
certain predictors of students' achievements in mathematics are of particular
interpretational interest. Through additional simulation studies and another
application to a well-known benchmark dataset, we also show competitive
performance when compared to regression models, alternative formulations of
semi-parametric BART, and other tree-based methods. The implementation of the
proposed method is available at \url{https://github.com/ebprado/CSP-BART}.
