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In general, you want to check the performance of basic machine learning algorithm in default settings to get a feeling for the underlying structures, before picking a model.

I know every ternary tree can be represented as a binary tree. But why is this a reason to not use ternary trees? Of course they are computationally much more expensive but that is the same for other algorithms. Also, you could use them if the number of features is low but the hypothesis space is huge.

Why not include ternary trees in the process of spot checking algorithms to see if they outperform binary trees?

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Instead of focusing so much on decision tree classifiers, consider looking at Leo Breiman's Random Forests classifier. Once you understand the elegance of this approach (bootstrapping to train trees, dropping test samples down trained trees for testing, shuffling feature values in trained trees to obtain importance scores, etc.), you will not likely care so much about binary vs ternary straightforward decision tree classifiers, CART, etc. The potential for overfitting is also lower with RF.

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