We consider in this paper a scheduling problem with small communication delays and an unbounded number of processors. It is known that duplication can improve the makespan of schedules. However, scheduling algorithms may create a huge amount of duplicates. The volume of a schedule is defined as the total number of tasks (i.e., original and duplicates). Assuming that the tasks have the same processing time d, that communication delays are all equal to c⩽d and that the precedence graph is an out-tree, we study the problem of finding the minimum volume of a feasible schedule with makespan t. We derive some dominance properties and prove that this problem is polynomial using a dynamic programming algorithm.