As a promising new technology for green communication, backscatter communication has attracted wide attention in academics and industry. This paper studies the resource allocation problem for an unmanned aerial vehicle (UAV)-assisted backscatter communication network. The UAV is used as an airborne mobile base station (BS) by time division multiple access (TDMA) protocol to help backscatter devices (BDs) forward signals to backscatter receiver (BR). Considering the fairness among BDs, we investigate maximizing the minimum (max-min) rate of the proposed network by jointly optimizing backscatter device scheduling, reflection coefficient, UAV's power control, and UAV's trajectory. The optimization problem is a non-convex problem, which is challenging to obtain the optimal solution. Therefore, we propose an efficient iterative algorithm to decompose the optimization problem into four subproblems by the block coordinate descent (BCD) method. The variables are alternatively optimized by the interior point method and successive convex approximation (SCA) techniques in each iteration. Finally, simulation results show that the max-min rate of the system obtained by the proposed scheme outperforms other benchmark schemes.