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I know that a decision tree recursively splits along each attribute, greedily minimizing the wrong classifications/deviance at each split.

But, what is the order in which the attributes are split? In other words, for a regression tree in N dimensions, what determines which attribute gets split first?

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One option is to use a simple approach, like just choosing random attributes, or taking one at a time.

Another option is to use a metric to decide, at each step, the attribute that best splits the data. Different tree algorithms apply different metrics, like for instance Gini impurity (CART algorithm), or information gain (ID3 and C4.5 algorithms).

You can read a brief description of these methods in the "Decision tree learning" entry on the Wikipedia.

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