1908 Edwardsville City Directory
George H Walby, miner, home at 23 Murray
I didn't find him in 1905. And the 1907 Directory of Wilkes-Barre did not include Edwardsville.
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There is a Sanborn insurance map of Edwardsville here, from about 1903:
1903 Map part 1
https://www.loc.gov/resource/g3824em.g3824em_g076451903/?sp=1&r=0.183,0.118,0.54,0.276,01903 Map part 2
https://www.loc.gov/resource/g3824em.g3824em_g076451903/?sp=2&r=0.011,-0.036,0.934,0.477,0I didn't find Murray Street on the map.
There is a Welsh Congregational Church on the corner of Church and Main, which could be the church Erato posted. It is on part one of the map.
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familysearch says that they have an index of Pennsylvania births from 1906 to 1913 here:
https://www.familysearch.org/search/collection/1681005?collectionNameFilter=falseI did not find Gertrude Walby, 1905 to 1910
The state claims to have birth records from 1906 here
https://www.health.pa.gov/topics/certificates/Pages/Birth-Certificates.aspxThey charge $20 and $10 for handling
I doubt they will refund your money if they don't find the birth.
Personally, I wouldn't risk the money. You already know her name, date of birth, and parents. There isn't much else it would tell you, except maybe George's street address and occupation. - which the City Directory tells you. And it isn't in the index, so maybe it wasn't registered, or the index doesn't include Edwardsville so maybe the state's index doesn't either, or Edwardsville didn't register the births, or didn't report them to the state, etc., etc.
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basic resources for the area, from the local library:
https://osterhout.info/using-the-library/genealogy-services/There is a chance there was a birth announcement in the paper. The library might have it. You could risk the $10 on it if you like. Be sure they know the family was Welsh. There might have been a Welsh language paper at the time, you never know.
And they might be willing to help you find the most likely church, if Dr Edwards' church is not the one. But I would bet first on the church being helpful, especially if it is still predominantly Welsh.