The study of bounds for topological indices based on various graph parameters, as well as the identification of extremal graphs where these bounds are achieved, is an active and intriguing area of research in graph theory. The Euler Sombor index (\(\mathcal{ESI}\)) is a recently introduced degree-based index with a strong linear relationship to several physicochemical properties of octanes. This work explores extremal trees of \(\mathcal{ESI}\) for given graph parameters. First, we identify the second smallest and second largest values of the Euler Sombor index in terms of the graph order \( n \). Then, we characterize maximal and minimal quasi-trees for given \(n\) employing the majorization principle. Finally, we identify the extremal trees for the \(\mathcal{ESI}\) index for given \(n\) and domination number.