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In this paper we consider a linearized variable-time-step two-step backwarddifferentiation formula (BDF2) scheme for solving nonlinear parabolicequations. The scheme is constructed by using the variable time-step BDF2 forthe linear term and a Newton linearized method for the nonlinear term in timecombining with a Galerkin finite element method (FEM) in space. We prove theunconditionally optimal error estimate of the proposed scheme under mildrestrictions on the ratio of adjacent time-steps, i.e. $0<r_k < r_{\max}\approx 4.8645$ and on the maximum time step. The proof involves the discreteorthogonal convolution (DOC) and discrete complementary convolution (DCC)kernels, and the error splitting approach. In addition, our analysis also showsthat the first level solution $u^1$ obtained by BDF1 (i.e. backward Eulerscheme) does not cause the loss of global accuracy of second order. Numericalexamples are provided to demonstrate our theoretical results.

Unconditionally optimal error estimate of a linearized variable-time-step BDF2 scheme for nonlinear parabolic equations