Alexander's first wife, Susan, died on 29 Sep 1924 so there was plenty of time for them to have children. However, I have not found any.
Hello...
I think they had at least two children.
Just browsing through the Ancestry electoral roll indexes and an Alexander Thomas Donald BARRON and a Susan Finnimore BARRON both appear in the Dunedin West (NZ) electorate in 1911 and then the Melbourne Ports (Australia) electorate in 1914. They must have then returned to NZ as he appears in the Egmont (NZ) electorate in 1919. There is also a Susan BARRON in the same electorate so, without looking at the actual roll, I'm guessing it's her.
The NZ WW1 Reserves Index (compiled between August and October 1917) lists an Alexander T D BARRON of Eltham, Second Reserves, occupation Salesman, classification C.
I'm guessing this is your Alexander as the initials are the same and I'm fairly sure Eltham is in the above mentioned Egmont electorate. Classification C means two children.
Will check the NZ BDM Birth fiches to see if there's are any BARRON births recorded in Eltham in the 1914-1917 period. Won't be till later in the week, unless someone else can do it sooner. Might be a lead to 2011. Might not :-)
Regards
Beg