Semiconductor tetrapods are three-dimensional (3D) branched nanostructures, representing a new class of materials for electrical conduction. We employ the single-electron transistor approach to investigate how charge carriers migrate through single nanoscale branch points of tetrapods. We find that carriers can delocalize across the branches or localize and hop between arms depending on their coupling strength. In addition, we demonstrate a new single-electron transistor operation scheme enabled by the multiple branched arms of a tetrapod: one arm can be used as a sensitive arm-gate to control the electrical transport through the whole system.