We propose an integrated behavior and motion planning framework for the lane-merging problem. The behavior planner combines search-based planning with game theory to model vehicle interactions and plan multivehicle trajectories. Inspired by human drivers, we model the lane-merging problem as a gap selection process and determine the appropriate gap by solving a matrix game. Moreover, we introduce a branch model predictive control (BMPC) framework to account for the uncertain equilibrium strategies adopted by the surrounding vehicles, including Nash and Stackelberg strategies. A tailored numerical solver is developed to enhance computational efficiency by exploiting the tree structure inherent in BMPC. Finally, we validate our proposed integrated planner using real traffic data and demonstrate its effectiveness in handling interactions in dense traffic scenarios.