linear regression vector of the parent node. In particular, if for a given leaf $s$ we have computed an optimal regularized least squares solution $w_s^*$; when further splitting this node into two (children) leaves we optimize problems of the form $ \min_{w_{i,k} \in \mathbb{R}^{d}} \paren{ \norm[1]{\bX_{A^{}_{i,k}}{w}^{}_{i,k}-Y_{A^{}_{i,k}}}_{P}^{2} + \lambda \| w_{i,k} - w_s^*\|^2_Q}$ and in the similar form for $A_{i,k}^{c}$. Thus, the solution at lower nodes are regularized by the propagation of the solutions from higher (parential) nodes. As will be shown in the empirical evaluation, such a regularization is crucial to the stability of the proposed algorithm. This was in fact a key insight of early experiments, without strong regularization of this form the proposed trees were not able to generalize well. We note that $L_{i,k}$ can be analytically computed by standard linear algebra formulas, see Section \[sec:ctn\] for a more detailed discussion of numerical aspects.

As the optimal empirical splitting rule, we choose a pair $(i^{\star},k^{\star})$ minimizing the above: $$(i^{\star},k^{\star}) = \argmin_{(i,k) \in [D] \times {\mathcal{J}}} L^\lambda_{i,k}.$$ For the pair $(i^{\star},k^{\star})$, we split the data accordingly and get the subsets $\paren{\bx_\ell,y_\ell}_{\ell \in {A_{i^{\star},k^{\star}}}}$ and $\paren{\bx_\ell,y_\ell}_{\ell \in {A^c_{i^{\star},k^{\star}}}}$ for the two children of the node. Now, provided that none of 