  The transmission of a vertex in a connected graph is the sum of its distances
to all the other vertices. A graph is transmission irregular (TI) when all of
its vertices have mutually distinct transmissions. In an earlier paper,
Al-Yakoob and Stevanovi\'c [Appl. Math. Comput. 380 (2020), 125257] gave the
full characterization of TI starlike trees with three branches. Here, we
improve these results by using a different approach to provide the complete
characterization of all TI starlike trees. Moreover, we find the precise
conditions under which a double starlike tree is TI. Finally, we implement the
aforementioned conditions in order to find several infinite families of TI
starlike trees and TI double starlike trees.
