The Burnet descent is fully described and referenced in the book The Omnibus Ancestry: 619 Documented American and European Lines. It is available for download through Lulu.
The patriarch of the Burnet family of Southampton, Long Island, NY, was Thomas Burnet, first known as a settler who arrived at Scout’s Bay, western Long Island, in 1640/1, and who died in Southampton in 1684 . My wife descends from him by way of her Gullett ancestors. Thomas’ great-great-great-granddaughter Jane Burnett (1820/1-1883) married in 1849, as her second husband, Joshua Gullett (1825-1863), in Washington county, Indiana .
For a number of years it has been claimed that Thomas of Southampton was the son of Thomas Burnet, a physician in Braintree, co. Essex, England, and grandson of Alexander Burnet, 11th laird of Leys, co. Kincardine, Scotland, a connection explicitly stated in E. A. Bailey’s Crannog to Castle (2000) . The laird, whose wife was Katherine Arbuthnot, has extensive traceable ancestry that includes a descent from King Robert III of Scotland .
Grounds for Skepticism
Much as I would like to trace a royal line of descent on behalf of my wife, I have been skeptical of the stated connection between the Southampton and Leys Burnetts. There are three major grounds for skepticism, as follows.
1. The reference given in Crannog to Castle for the Southampton to Leys connection was the “personal research notes” of a private individual. In absence of a description of the evidence, such notes cannot be assumed authoritative.
2. There is little similarity between the names of the children of Thomas of Southampton (John, Lot, Hester, Joel, Aaron, Miriam, Priscilla, Daniel, Lois, Mordecai, and Matthias) and those of the children of Thomas of Braintree (Thomas, Alexander, Frances), his wife (Jane), or his siblings (Alexander, Robert, Gilbert, Duncan, John, Janet, Margaret, and Elizabeth) . It seems an important observation that the children’s names in the Southampton family appear to be solidly English, while several names in the Braintree family are commonly Scottish. But in any case, the lack of resemblance between the two families is a reason to doubt that one closely derived from the other.
3. There is also some question whether the chronology is consistent. The Braintree man did have a son named Thomas, who was b. 1612 . However, the second child named in the Southampton man’s will, Lot, was m. 1675, and the third child, Hester, married a man b. 1649 . These dates suggest that the first child, John (for whom no firm birth or marriage dates are otherwise known) was b. ca 1648? . In turn that is suggestive that his father Thomas was born later than 1612, possibly closer to 1622.
While none of these points is definitive, together they provide ground for skepticism about the Southampton-to-Leys connection, at least as it has been described.
Support from Y-DNA Testing
A fly in the skepticism ointment, however, is that Y-DNA testing does in fact support a connection between the Southampton and Leys families. Y-DNA comparisons have been made between two descendants of Thomas of Southampton on the one hand, and on the other hand, a descendant of Alexander Burnet, 11th laird of Leys (b. ca 1540?, d. 1578) and a descendant of Alexander Burnet, 9th laird of Leys (b. 1500, d. 1574). The former match the latter in either 32/37 or 62/67 markers depending on the pairing .
These results appear definitive, establishing that there is in fact a genetic connection between Thomas of Southampton and the Burnett lairds of Leys. But is the connection through the 11th laird? The results indicate, on average across the outcomes, a 50% probability of convergence in 15.5 generations . At 31 years a generation  and assuming an average birthyear of 1970 across the testees, that would indicate a common ancestor b. ca 1490? — i.e., more likely in the person of the 9th than the 11th laird. However, the error margin is very high, and the results are consistent with either a much earlier or a much later convergence, including one with the 11th laird. Hopefully estimates will narrow as more tests are received.
Others of the Name
A final consideration is that there was no shortage of persons named Thomas Burnett in British records of the early 17th century. Any of several could have been the American immigrant, instead of the son of Thomas of Braintree. Thus those of the name born 1617-1627 in England include Thomas Burnett, chr. 1620 in Stepney, London, Eng, son of John and Elizabeth; Thomas Burnett, chr. and d. 1621 in Lincoln, co. Lincoln, son of John; Thomas Burnet, chr. 1622 in Fincham St. Martin with St. Michael, co. Norfolk, Eng, son of John; Thoms. Burnet, chr. 1623 in Otley, co. York, son of Thoms.; and Thomas Burnet, chr. 1625 in Adstock, co. Buckingham, son of Richard . This list, of course, is only as complete as the church records from which it was compiled, and there were others of somewhat different spellings of the surname.
Earlier, there were the intriguing christenings of Mathias Burnet (1594) and Joell Burnet (1599), both sons of Thomae (Thomas) Burnet, in Holbeach, co. Lincoln . There was also a Danyell Burnett m. 1631 in Holbeach, whose baptism was not recorded there , but who could have been another son of Thomas. Mathias, Joel, and Daniel are three of the unusual names in the Southampton family, possibly suggesting a relationship. However Tho. [Thomas], the apparent brother of Mathias and Joell (and possibly of Danyell), was chr. 1609 , raising the same chronological issue as the Leys connection.
That the ancestry of Thomas Burnet of Southampton, Long Island, ultimately traces to the Burnetts of Leys in Scotland appears indisputable, given the Y-DNA evidence. However, there remain grounds for skepticism over whether the relationship was as close as a descent from the 11th laird. If the line instead went through one of the other English families of record, such as that of Thomas of Holbeach, co. Lincoln, there could have been any number of connecting generations before convergence with the Leys family.
Unfortunately, for now that keeps me from claiming a royal descent for my wife . But perhaps it offers the prospect of one, following some future application of the genealogist’s craft.
 He was not, as frequently asserted, previously of Lynn, Massachusetts. See The Omnibus Ancestry for a discussion.
 Boles, D.B. (2017). The Omnibus Ancestry: 619 Documented American and European Lines, 3rd ed. Lulu e-books.
 Leys is near Tarland, co. Kincardine. Although it was the original seat of the family, in the 16th century the Burnetts built Crathes Castle in co. Aberdeen as their home. Crathes Castle is about 23 road miles ESE of Leys. There was an earlier castle at Leys, located on a crannog (an artificial island) that today is almost inaccessibly surrounded by marsh. Its foundations were destroyed during a 19th-century excavation. It was reputedly the home of the Burnetts from 1323 to 1550 (information retrieved from https://canmore.org.uk/site/36684/loch-of-leys, 2019).
 Information retrieved from http://www.thepeerage.com/p19019.htm#i190189 & linked pages (2019).
 Information retrieved from https://en.wikipedia.org/wiki/Crathes_Castle (2018).
 Crannog to Castle stated that he was b. 1645, but there appears to be no basis for the assertion in primary records. My estimate is based on the suppositions that Hester was about 4 years younger than her husband and was b. ca 1653?; Lot was about 25 years old at the time of marriage and was b. ca 1650?; and John was about 2 years older than Lot and was b. ca 1648?.
 Calculator at https://clandonalddnaproject.org/index.php/tmrca-calculator (2017).
 FamilySearch, LDS church (2019).
 If the ancestry of my wife, along with that of the Southampton-descending Y-DNA testees, converges with the Leys line anytime beginning with the 9th laird’s grandfather or later, they would be descendants of King Robert III. The 9th laird’s grandfather, also named Alexander Burnett, had his children by his second wife Marjory Forbes. Her ancestor James, 2nd Lord Forbes, was the son of a granddaughter of Robert III (www.thepeerage.com, op. cit.).
Modification of “Crathes Castle (XVIe si\xe8cle), Banchory, Aberdeenshire, Ecosse, Grande-Bretagne, Royaume-Uni” (https://ccsearch.creativecommons.org/photos/d8c14647-6663-4132-9def-84f4e02142b7) by byb64 (en voyage jusqu’au 26) (https://www.flickr.com/photos/50879678@N03), licensed under CC BY-NC-SA 2.0 (https://creativecommons.org/licenses/by-nc-sa/2.0).