A note on extremal trees with degree conditions

A fundamental question in the study of graph invariants asks for the extremal structures under certain constraints that maximize or minimize a graph invariant. In this note, we summarize some recent work on the extremal trees of distance-based and degree-based graph invariants under various degree conditions. We note that many of such extremal structures turned out to be identical for different but similar invariants. Such common extremal structures are investigated through the greedy trees and majorization between degree sequences. We show that many of the known extremal results can be obtained through this line of arguments. We also introduce some new extremal results as immediate consequences.