The general Sombor ( \(\mathcal{S}\mathcal{O}_\alpha \) ) index of a graph G is defined as the sum of weights \(\Big (d^2_x(G) +d^2_y(G)\Big )^\alpha \) over all edges xy of G, where \(\alpha \ne 0\) is a real number and \(d_x(G)\) denotes the degree of a vertex x in G. In this paper, we focus on two specific classes of trees: \({{\mathcal {T}}}_{n,b}\) , the set of all n-vertex trees with b branching vertices, and \({{\mathcal {T}}}_{n,\Delta }\) , the set of all n-vertex trees with prescribed maximum degree \(\Delta \) . Thus the purpose of this paper is twofold concerning the \(\mathcal{S}\mathcal{O}_\alpha \) index: (i) to characterize the minimal trees in \({{\mathcal {T}}}_{n,b}\) when \(\alpha > 0\) , and (ii) to characterize the maximal trees in \({{\mathcal {T}}}_{n,\Delta }\) when \(0<\alpha < 1\) . The results of (i) hold true even when the class \({{\mathcal {T}}}_{n,b}\) is confined to the class of chemical trees and also recover previously known results for the Sombor index. The findings in (ii) resolve a previously posed problem for the \(\mathcal{S}\mathcal{O}_{\alpha }\,(0<\alpha <1)\) index and, moreover, establish analogous results for the well-known general sum-connectivity index, thereby addressing the corresponding unresolved cases for both indices.
