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Behavior Trees (BTs) have attracted much attention in the robotics field in recent years, which generalize existing control architectures and bring unique advantages for building robot systems. Automated synthesis of BTs can reduce human workload and build behavior models for complex tasks beyond the ability of human design, but theoretical studies are almost missing in existing methods because it is difficult to conduct formal analysis with the classic BT representations. As a result, they may fail in tasks that are actually solvable. This paper proposes BT expansion, an automated planning approach to building intelligent robot behaviors with BTs, and proves the soundness and completeness through the state-space formulation of BTs. The advantages of blended reactive planning and acting are formally discussed through the region of attraction of BTs, by which robots with BT expansion are robust to any resolvable external disturbances. Experiments with a mobile manipulator and test sets are simulated to validate the effectiveness and efficiency, where the proposed algorithm surpasses the baseline by virtue of its soundness and completeness. To the best of our knowledge, it is the first time to leverage the state-space formulation to synthesize BTs with a complete theoretical basis.