We give a way to obtain formulas for ${pi}*{psi}^{kappa}_{n+1}$ in terms ${psi}$ and ${lambda}-classes$ where ${pi}=ar M_{g,n+1}{ ightarrow}ar M_{g,n}(g=0,;1,;2)$ by the localization theorem. By using the formulas, we obtain Kontsevich-Manin type reconstruction theorems for $ar M_{0,;n}(mathbb{R^m}),;ar M_{1,;n},;and;ar M_{2,;n}$. We also (re)produce a lot of well-known relations in tautological rings, such as WDVV equation, the Mumford relations, the string and dilaton equations (g = 0, 1, 2) etc. and new formulas for ${pi}*({lambda}_g{psi}^{kappa}_{n+1}+...+{psi}^{g+{kappa}_{n+1}$