CART stands for "classification and regression trees". This means CART would refer to any tree which has its complexity optimized by (in other words pruned using) misclassification rate or average squared error (ASE). Given this, then the "probability tree" would be considered as a CART tree.
CHAID, ID3 and C4.5 are all GROWTH options, meaning they control how the tree is grown, in other words how splits are determined. So it would need to be stated how the trees were grown, before one could say if they are CHAID, ID3 or C4.5. So technically any of the tree names you asked about, probability tree, maximal tree, pruned tree, could each be any one of CHAID, ID3, or C4.5. Specifically to the AAEM course, trees are build using only logworth, which is CHAID. So in the context of AAEM, you could say all 3 of the trees you asked about were grown using logworth (based on Chi-squared) so they could be called CHAID trees. Outside the AAEM class, however, one would need to know how the tree was grown.
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