CART algorithm (Classification and regression trees) question
Making an additional split will always decrease $Err$ (until pure leaves are reached), so any negative $\alpha$ will have the same minimizing $T$ as $\alpha=0$.  So we might as well take $\alpha\geq0$.  (But there is no requirement that $\alpha$ be an integer.)

Erm, unless you implement things so that even a pure node can be split, in which case a negative $\alpha$ will make the tree split even these, until every leaf consists of a single sample (and duplicates thereof) or the maximum depth is reached.