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For example, let's say that the age (say $x$) of a person for $12< x< 25$ can be used to predict computer usage to a high degree of certainty. In a decision tree, this could be represented by a split of $x<25$ followed by a split of $x>12$.

Does xgboost, or other decision tree learning algorithms in general, handle multiple splits within the same branch?

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Yes, it can handle multiple splits withing the same branch.
A decision tree model can use the same feature as many times as optimally needed.
You can see an example in the sklearn doc.
The petal-length feature is used multiple times in the same branch.

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