Many definitive and approximate methods have been so far proposed for the construction of an optimal binary search tree. One such method is the use of evolutionary algorithms with satisfactorily improved cost efficiencies. This paper will propose a new genetic algorithm for making a near-optimal binary search tree. In this algorithm, a new greedy method is used for the crossover of chromosomes while a new way is also developed for inducing mutation in them. Practical results show a rapid and desirable convergence towards the near-optimal solution. The use of a heuristic to create not so costly chromosomes as the first offspring, the greediness of the crossover, and the application of elitism in the selection of future generation chromosomes are the most important factors leading to near-optimal solutions by the algorithm at desirably high speeds. Due to the practical results, increasing problem size does not cause any considerable difference between the solution obtained from the algorithm and exact solution.