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Suppose we have a set of objects, each belonging to one of the disjoint categories $c_1, ... , c_n$. Suppose further that for every single category $c_i$ there is a corresponding set $T_i$ that contains all optimal binary classification trees that only split the objects based on whether they belong to $c_i$ or not. Optimal is defined as having the least sum of distances from the root to the leaf nodes.

Question: Can the sets $T_1, ..., T_n$ be used effectively to construct a single global binary classification tree that classifies objects for every category and is also optimal?

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