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 $alphageq0$. (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.

Or does the algorithm state that alpha must be a non negative integer at all times?

As far as I know that is actually the case:

In "The Elements of Statistical Learning" (1) on page 308 the authors write:

The tuning parameter $alpha geq 0$ governs the tradeoff between tree size and its goodness of fit to the data.

Also, the scikit-learn documentation says:

Minimal cost-complexity pruning is an algorithm used to prune a tree to avoid over-fitting, described in Chapter 3 of [BRE]. This algorithm is parameterized by $alpha geq 0$ known as the complexity parameter.

Both sources refer to "Classification and Regression Trees" (2) as the original source.

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(1) "The Elements of Statistical Learning"; Hastie et al; 2nd Ed; 2008

(2) "Classification and Regression Trees"; Breiman, et al; 1984

? is a parameter used for regularization which is controlling the tree from being overfit , an inherent issue with CART algorithms. A negative ? would lead to a larger tree which may accentuate the overfitting issue