Solving the optimal classification tree problem enables to compute classifiers which are both interpretable and efficient. Most of the exact methods for this problem are based on on a Mixed Integer Linear Program (MILP) formulation. However, the efficiency of MILP solvers generally does not allow these formulations to be solved directly, once the dataset exceeds a critical size. To address this challenge, we propose in this paper an iterative exact algorithm than handles medium-sized datasets from the state-of-the-art. The basic idea is to start by solving a MILP formulation on a small subset of data points representative of the considered dataset. Then, the subset is iteratively extended until global optimality of the initial problem is reached. A key feature is to compute relevant initial subsets of data points. For this, we introduce the concept of data-partitions and design several algorithms to compute them. We then define two MILP formulations to compute optimal classification trees on data-partitions. We prove that combining our iterative algorithm with our first formulation enables to obtain an optimal solution of the original problem. We also propose an alternative method based on the second formulation which is significantly faster. We present extensive computational experiments to compare our algorithms with state-of-the-art approaches. We show that our methods constitute the best compromise between in-sample accuracy and interpretability.