Mary Allen, daughter of Increase and Rachel Allen, was born May 29, 1689 in Dartmouth, Bristol County, Massachusetts. Her father, Increase Allen was a Quaker who, among other things, in Dartmouth on 4 Jan 1696/7 helped to "Inventory the estate of William Wood, Sr., of Dartmouth late dec'd taken 4 Jan 1696..." 
Her grandfather's brother, great-uncle William Allen, was a "prominent Friend who with Edward Wanton, Joseph Coleman and Nathaniel Fitsrandal, (among many other activities) presented to the General Court of Plymouth 'conscientiously and in all tenderness' their reasons why they could not 'give maintenance to the established preachers." Thus giving 'voice' to a common objection of these early Quakers both in the new colonies and in England.
From Dartmouth, MA., to Warwick, R.I.: Marriage
Currently located are 2 pertinent marriage records records for two Society of Friends (colonist) record-keeping locations: (1) Mary Allen married John Green on 16d 12m 1709 [February 16, 1710] in Dartmouth, Bristol County, Massachusetts Bay Colony. Traveling on land, Dartmouth, MA is about 42 miles distance to Warwick. 'As-the-crow-flies' across the waters the distance is 8 miles.
(2) Mary Allen, daughter of Increase Allen, married 16 Feb 1709/10 at Warwick, Rhode Island, John Greene, son of James Greene. 
John & Mary (Allen) Greene had the following children recorded in the Narragansett Friends Records which suggests husband John Greene was also a Quaker or had converted. These children were all born at Potowomut according to this record: 
David Greene, b. 4d 11th m 1710/1, parents John & Mary Greene
James Greene, b. 19d 1m 1713, parents John & Mary
Rachel Greene, b. 16d 1m 1714-15, parents John & Mary; m. Francis Matteson
James / Increase (may be Increase, see children listing below) Greene, b.12d 2m 1717, parents John & Mary
Elizabeth, b. 26d 3m 1719, parents John & Mary
A listing of children is also recorded in Warwick, Kent county, Rhode Island state vital records which includes the children recorded in the Narragansett Quaker record and additionally picks up where that record leaves off beginning with Benjamin. They were all born in Warwick. Warwick is about 20 miles southwest of Potowomut. The record reads: "Of John and Mary Greene" 
David, b. 4 Jan 1710/11
James, b. 14 Mar 1712/3; d. 30 May 1792, (West) Warwick, R.I.
Rachel, m. Francis Matteson; d. 8 Mar 1740, Warwick, R.I.
Increase, b. 12 Apr 1717
Elizabeth, b. 26 May 1719
Benjamin, b. 26 Sep 1721
Dina, b. 5 Jan 1723/4
Mary, b. 1 Jan 1724/5; d. 13 Oct 1727
Joseph, b. 19 Feb 1727/8
Peter, b. 8 Jan 1730/1
Martha Greene, dau of John Greene, m. Joseph Matteson, son of Thomas Matteson, 8 May 1727 in West Greenwich, Kent County, Rhode Island. They were married by Thomas Spencer, Justice. She is reported to have d. 1 Feb 1746 and buried in the Matteson Lot cemetery, West Greenwich, Kent, R.I. West Greenwich is about 5 miles distance from Warwick.
Common Name Caution
Mary, dau of Incease Allen not to be confused with Allen-407 Mary Allen (1688-1766) born in Salem, MA, dau of Joshua and Mary (Crowell) Allen; m. Windham, CT., Ebenezer Abbe (1683-1758); d. Windham, CT.
There are numerous Mary Allen (LNAB & married name) in New England about the same time.
↑ Rhode Island, Vital Extracts, 1636-1899; Vol.01/21; image pg.169; Warwick BMDs: Kent County. Typed ledger Reprinted by Ancestry.com Operations, Inc., 2014, Provo, UT, USA. Online db access 20 Jan 2020.
↑ Rhode Island, Vital Extracts, 1636-1899; Vol. 01: Kent County, BMD; Vol.1; image pg.47. Reprinted by Ancestry.com Operations, Inc., 2014, Provo, UT, USA. Online db access 20 Jan 2020.
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It may be possible to confirm family relationships with Mary by comparing test results with other carriers of her mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share DNA with Mary: