We consider an unmanned aerial vehicle (UAV) assisting data collection from multiple sensor nodes (SNs). We provide a completion time minimization design via jointly deciding the UAV trajectory and the SN assignment scheme. In particular, we first characterize the fundamental features of the joint optimal solution to the formulated problem. On the one hand, the optimal UAV trajectory is proved following a successive-hover-fly (SHF) structure. Namely, in an optimal solution, the UAV successively visits multiple hovering points and performs hovering with designated duration, while the maximum speed is achieved during the whole flying period between each two hovering points. On the other hand, the optimal SN assignment is characterized to follow a segment-based scheme. Based on the two characterizations, we are motivated to implement SHF structure with turning points in trajectory design and reasonably assume each segment in SHF structure having constant SN assignment. Afterwards, we relax the binary constraints for SN assignments and establish a convex approximation for the reformulated problem, which enables an iterative algorithm. A suboptimal joint solution is obtained via iteratively optimizing the completion time. A realization strategy is also provided for the relaxed solution while assuring the completion of data collection tasks. Finally, the proposed solution is validated and evaluated through numerical results. Both a low complexity and an accurate task completion guarantee of our proposed solution are observed in comparison with the benchmarks.