Hi, Danette,
I have run through several ways of searching for Emma and Gertrude, George and/or Charles Button, and any permutations thereof. Given the sparse information provided on census records and the fact that Emma would have been only 15 years old when Gertrude was born, I suspect there was no Mr. Button and Button was probably not Emma's maiden name either. I think it is probable that Gertrude was illegitimate and places of birth for both her and her mother may or may not be correct. Depending on your level of interest in tracing your roots, you may want to consider DNA testing.