# Can an optimal global classification tree be constructed from trees for single categories?

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?