A ‎note‎ ‎on‎ ‎the‎ ‎r‎e-defined third Zagreb index of trees

For a graph $\Gamma$‎, ‎the re-defined third Zagreb index is defined as $$ReZG_3(\Gamma)=\sum_{ab\in E(\Gamma)}\deg_\Gamma(a) ‎\deg_\Gamma(b)\Big(‎\deg_\Gamma(a)+‎\deg_\Gamma(b)\Big)‎‎,$$‎‎where $\deg_\Gamma(a)$ is the degree of‎ ‎vertex $a$‎. ‎We prove for any tree $T$ with $n$ vertices and maximum degree $\Delta$‎, ‎‎$ReZG_3(T)\geq‎16n+\Delta^3+2\Delta^2-13\Delta-26$ ‎when ‎‎$‎\Delta n-1‎$ ‎and‎ $ReZG_3(T)=‎n\Delta^2+n\Delta-\Delta^2-\Delta$ ‎when ‎‎$‎\Delta=n-1‎$. ‎Also we determine the corresponding extremal trees‎. ‎‎