An unmanned aerial vehicle (UAV) has been considered as a base station (BS) or a relay node in wireless communication systems due to its cost-effective and fast deployment. UAV BSs (UBSs) can provide a wireless connectivity to user equipments (UEs) within a given coverage, while UAV relay systems (URSs) can extend a service coverage of BS to provide a reliable communication link to an isolated UE (IU). The deployment of UAV(s) impacts greatly on the performance of UBS and URS. In this paper, a novel UAV deployment algorithm for multi-layer URS is proposed to maximize an average data rate of UEs while guaranteeing a seamless communication service to IU. The proposed algorithm utilizes the minimum number of UAVs and derives the minimum number of transmission time slots to improve an average data rate. In addition, a single-layer URS is considered as a special case of multi-layer URS, and its deployment algorithm is introduced. The performance comparison between multi-layer and single-layer URSs is presented in numerical results based on the position of IU.