On Extremal Family Trees $(\mathcal{T}_n)_{n\geqslant 3}$ Beyond Caterpillars and Greedy Constructions

This paper investigates topological indices for the greedy tree $\mathcal{T}_\mathscr{D}$ associated with a graphic degree sequence $\mathscr{D} = (d_1 \geqslant d_2 \geqslant \dots \geqslant d_n)$ of a tree. A fundamental challenge in the study of topological indices lies in establishing precise bounds, as such findings illuminate intrinsic relationships among diverse indices. We investigate the extremal properties of the graph invariant $σ$ over the family $\mathcal{T}_n$ of all trees on $n \ge 3$ vertices. Specifically, we compare the minimum values of $σ$ attained in restricted subclasses -- including caterpillar trees and greedy trees -- with the global minimum over $\mathcal{T}_n$. We prove that caterpillar trees do not achieve the minimum value of $σ$ among all trees, whereas greedy trees attain values no smaller than this global minimum. Moreover, we show that certain trees, which are neither caterpillars nor greedy trees, have $σ$-values strictly between the global minimum over $\mathcal{T}_n$ and the minimum among caterpillar trees. These results highlight structural limitations of these common tree classes in extremal problems and offer new insights into the role of non-caterpillar, non-greedy trees in minimizing graph invariants.