Performing a deterministic reduction operation in a parallel computer that includes compute nodes, each of which includes computer processors and a CAU (Collectives Acceleration Unit) that couples computer processors to one another for data communications, including organizing processors and a CAU into a branched tree topology in which the CAU is a root and the processors are children; receiving, from each of the processors in any order, dummy contribution data, where each processor is restricted from sending any other data to the root CAU prior to receiving an acknowledgement of receipt from the root CAU; sending, by the root CAU to the processors in the branched tree topology, in a predefined order, acknowledgements of receipt of the dummy contribution data; receiving, by the root CAU from the processors in the predefined order, the processors' contribution data to the reduction operation; and reducing, by the root CAU, the processors' contribution data.