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The structural and elastic properties of TlN and AlN in both rock salt (cubic) and wurtzite (hexagonal) structures have been studied by first-principles calculations within the generalized gradient approximation. An efficient strain-stress method is employed to calculate the single crystal elastic stiffness constants. In addition, the elastic properties of polycrystalline aggregates including bulk modulus (B), shear modulus (G), Poisson's ratio, and anisotropy ratio are also determined and compared with the experimental and theoretical results available in the literature. It is found that the structure transition from rock salt to wurtzite occurs at 13.3 GPa for AlN and -21.0 GPa for TiN at 0 K. The predicted elastic stiffness constants decrease with increasing volume except for the c(44) of the wurtzite structure. Based on the calculated B/G ratios, we predict the ductile behavior for wurtzite TiN and the brittle nature for the others, i.e. rock salt TiN, rock salt AlN, and wurtzite AlN. We also find that rock salt TiN and wurtzite AlN are isotropic, while wurtzite TiN and rock salt AlN are anisotropic.