I have seen it both ways, but Trothen seems to be the most used.Bryan Trothen states,
"Trothen seems to be an entirely North American version of the name. In Devonshire it is Trathen. I have also seen Trathin and Treathen-probably variables. It may derive from 'troth' or 'truth' - meaning true loyalty or faithfulness. The name seems to be associated with villages and cities in north Devon - Bradworthy, Woolfardisworthy, Hartland, and Bideford. Buckland, Brewer are also mentioned; but I think the best name is Bitworthy Cottage, the house where Thomas and Mary Trothen lived in the first half of the last century (1800's)."
Trothen is for my grandparent's ancestors and I am researching that line.Any information is welcome.