Robots executing tasks following human instructions in domestic or industrial environments essentially require both adaptability and reliability. Behavior Tree (BT) emerges as an appropriate control architecture for these scenarios due to its modularity and reactivity. Existing BT generation methods, however, either do not involve interpreting natural language or cannot theoretically guarantee the BTs' success. This paper proposes a two-stage framework for BT generation, which first employs large language models (LLMs) to interpret goals from high-level instructions, then constructs an efficient goal-specific BT through the Optimal Behavior Tree Expansion Algorithm (OBTEA). We represent goals as well-formed formulas in first-order logic, effectively bridging intent understanding and optimal behavior planning. Experiments in the service robot validate the proficiency of LLMs in producing grammatically correct and accurately interpreted goals, demonstrate OBTEA's superiority over the baseline BT Expansion algorithm in various metrics, and finally confirm the practical deployability of our framework. The project website is https://dids-ei.github.io/Project/LLM-OBTEA.