Semantic maps represent the environment using a set of semantically meaningful objects. This representation is storage-efficient, less ambiguous, and more informative, thus facilitating large-scale autonomy and the acquisition of actionable information in highly unstructured, GPS-denied environments. In this letter, we propose an integrated system that can perform large-scale autonomous flights and real-time semantic mapping in challenging under-canopy environments. We detect and model tree trunks and ground planes from LiDAR data, which are associated across scans and used to constrain robot poses as well as tree trunk models. The autonomous navigation module utilizes a multi-level planning and mapping framework and computes dynamically feasible trajectories that lead the UAV to build a semantic map of the user-defined region of interest in a computationally and storage efficient manner. A drift-compensation mechanism is designed to minimize the odometry drift using semantic SLAM outputs in real time, while maintaining planner optimality and controller stability. This leads the UAV to execute its mission accurately and safely at scale.