Johnathan Avery was born in Gilmanton, Strafford county, New Hampshire, the son of Josiah and Anna Avery. His father is honored by the Society of the Daughters of the American Revolultion for his Patriotic Service, as DAR Ancestor #A003995.
Jonathan Avery married Dorothy Dudley 16 Dec 1784 in Belmont county, New Hampshire.  Note: Belmont county was actually created from Strafford county, in 1840.
Applications for membership in the Society of the DAR have been made by the descendants of Johnathan Avery and Dorothy Dudley. 
Josiah Avery, born Oct 24, 1740, died Nov 25, 1801, is honored by the Daughters of the American Revolution. He signed the Association Test. 
↑ "New Hampshire Marriage Records, 1637-1947," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:FLX7-F2Z : 12 December 2014), Jonathan Avery and Dorothy Dudley, 16 Dec 1784; citing , Bureau of Vital Records and Health Statistics, Concord; FHL microfilm 1,001,121
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It may be possible to confirm family relationships with Jonathan 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 Jonathan: