The first Zagreb index $M_1$ of a graph $G$ is equal to the sum of squares of the vertex degrees of $G$. In a recent work [Goubko, MATCH Commun. Math. Comput. Chem. \textbf{71} (2014), 33--46], it was shown that for a tree with $n_1$ pendent vertices, the inequality $M_1\geq9n_1-16$ holds. We now provide an alternative proof of this relation, and characterize the trees for which the equality holds.