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DNA Connections
It may be possible to confirm family relationships with David by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with David:
There appears to be confusion on many BECK trees re William ALEXANDER & William Alexander BECK. As far as I can tell, William ALEXANDER is a separate individual referred to by Zella Armstrong in "History of Hamilton County, Tennessee" Vol 1 p 192. The article discusses a Wm ALEXANDER, Rev patriot, who died in 1838 at his son (David ALEXANDER) home in Dallas, Hamilton Co. TN followed (next par) by an article about David BECK, of the Dallas/Riverview area of Hamilton Co, same time period. Both families originated in PA, had ties to VA, and ended up in TN. I see no evidence that it was anything other than two separate families. Somehow, William Alexander has been connected to David BECK as his father & the entire ALEXANDER family line thus incorporated into the David BECK genealogy.