The CART stands for Classification and Regression Trees is a greedy algorithm that greedily searches for an optimum split at the top level, then repeats the same process at each of the subsequent levels.
Moreover, it does verify whether the split will lead to the lowest impurity or not as well as the solution provided by the greedy algorithm is not guaranteed to be optimal, it often produces a solution that’s reasonably good since finding the optimal Tree is an NP-Complete problem that requires exponential time complexity .
As a result, it makes the problem intractable even for small training sets. This is why we must go for a “reasonably good” solution instead of an optimal solution.