The recent evolution of ambient backscattering technology has the potential to provide long-range and low-power wireless communications. In this work, we study the unmanned aerial vehicle (UAV)-assisted backscatter networks where the UAV acts both as a mobile power transmitter and as an information collector. We aim to maximize the number of successfully decoded bits in the uplink while minimizing the UAV's flight time by optimizing its altitude. Power-domain NOMA scheme is employed in the uplink. An optimization framework is presented to identify the trade-off between numerous network parameters, such as UAV's altitude, number of backscatter devices, and backscatter coefficients. Numerical results show that an optimal altitude is computable for various network setups and that the impact of backscattering reflection coefficients on the maximum network throughput is significant. Based on this optimal altitude, we also show that an optimal trajectory plan is achievable.