This paper considers a high-rate turbo code which employs high-rate convolutional codes as component codes, and presents a novel method of reducing the decoding complexity of the codes. By eliminating some of branches that have the lowest reliabilities among all the branches entering each node, the proposed algorithm reduces the complexity in the process of the add-compare-select (ACS) between the consecutive stages of iterative decoding. That is, the complexity gradually decreases as the number of iterations increases. We compare the unpunctured high-rate turbo code with a classical punctured high-rate turbo code in terms of performance/complexity trade-off under the same code rate. Simulation results show that the proposed approach with a good trade-off provides an alternative coding scheme to the classical punctured high-rate turbo coding for the application to high-data-rate wireless communication systems.