The Stochastic Context Tree (SCOT) is a useful tool for studying infinite random sequences generated by an m-Markov Chain (m-MC). It captures the phenomenon that the probability distribution of the next state sometimes depends on less than m of the preceding states. This allows compressing the information needed to describe an m-MC. The SCOT construction has been earlier used under various names: VLMC, VOMC, PST, CTW. In this paper we study the possibility of reducing the m-MC to a 1-MC on the leaves of the SCOT. Such context trees are called perfect-memory. We give various combinatorial characterizations of perfect-memory context trees and an efficient algorithm to find the minimal perfect-memory extension of a SCOT.