A note on two characterizations on domination in the subdivision of trees

A set [Formula: see text] is a dominating set of a graph [Formula: see text] if every vertex of [Formula: see text] is adjacent to some vertex in [Formula: see text]. The domination number [Formula: see text] is the minimum cardinality among all dominating sets of [Formula: see text]. The subdivision graph [Formula: see text] of a graph [Formula: see text] is a graph obtained from [Formula: see text] by replacing each edge [Formula: see text] of [Formula: see text] by a vertex [Formula: see text] and edges [Formula: see text] and [Formula: see text]. [E. De Jesús Sánchez et al. On the (total) domination in subdivision graphs, Discret. Appl. Math. 384 (2026) 201–210] proved that for every tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, [Formula: see text]. In this paper, we provide a constructive characterization of extremal trees attaining these two bounds.