When growing a decision tree, we split the input space in a greedy, top$-$down, recursive manner. Given a parent region Rp$,$ we can choose a split sp$($j$,$ t$)$ which yields two child regions R$1=\{$X $|$ xj $<$ t$,$ X in Rp$\}$ and R$2=\{$X $|$ xj $$ t$,$ X in Rp$\}.$ Assuming we have defined a per region loss L$($R$),$ at each branch we select the