In this paper, the analytical bifurcation trees of period-1 motions to chaos in the periodically forced, quadratic nonlinear oscillator are discussed from the generalized harmonic balance method. The analytical solutions for stable and unstable periodic motions in such quadratic nonlinear oscillator are achieved, and the corresponding stability and bifurcation was discussed. The rich dynamical behaviors in such quadratic nonlinear oscillator are discovered, and this investigation help one better understand such complex dynamical behaviors. The analytical bifurcation trees from period-1 motion to period-4 motions in such quadratic oscillator are presented, and numerical illustrations of stable and unstable periodic motions are carried out by the numerical and analytical solutions. This investigation provides a comprehensive picture of complex periodic motion in the periodically excited, quadratic nonlinear oscillator.