Classification and Regression trees (CARTs) are prediction models based on local estimators assigned to non-overlapping subsets of the data space. To compute a CART model, splitting a hyperrectangular set into two subsets is a recurrent operation defined by the axis to split and the splitting proportion. The optimal axis and proportion should minimise the squared difference between the observed data and the local estimators. In optimisation terms, for any given axis, the squared difference is an objective function of the splitting proportion. In this paper, a Bayesian optimisation technique is applied to this objective function by means of a bound estimation based on a non-informative prior. The estimated bound is used as an acquisition function to search the optimal splitting proportion. Under challenging conditions, several Monte Carlo simulations show that this approach is statistically more accurate and robust in high dimensions than regular CART models.