Cases in point:
1.) President Chester A. Arthur
2.) Vice President Spiro Agnew.
President Chester A. Arthur's father, William Arthur, was born in County Antrim, Ireland. Chester A. Arthur was born in 1829 but his father was not naturalized a U.S. citizen until 1843.
Vice President Spiro Agnew's father, Theodore Spiros Anagnostopoulos, was born in Greece. Spiro Agnew was born in 1918 but his father was still listed as an "alien" in the 1920 U.S. Census.
In Inglis v. Trustees of Sailor's Snug Harbor, 28 U.S. 99 (1830), U.S. Supreme Court Justice Joseph Story wrote:
"Nothing is better settled at the common law than the doctrine that the children even of aliens born in a country while the parents are resident there under the protection of the government and owing a temporary allegiance thereto are subjects by birth."
In Inglis v. Trustees of Sailor's Snug Harbor, the Supreme Court rule that children born in New York City between July 4, 1776 and September 15, 1776 were U.S. citizens by birth but those children born in New York City between September 16, 1776 and November 24, 1783 were not U.S. citizens by birth.
Why?
Because, from September 16, 1776 to November 24, 1783, New York City was occupied by the British.
What mattered was that "the parents are resident there under the protection of the government and owing a temporary allegiance". After September 15, 1776, that "Government" in New York City was Great Britain. Therefore, the Supreme court ruled, any child born in New York City during the British occupation was a British subject by birth and not an American citizen by birth.
The 1920 Census is a mistaken entry. The census taker even has his wife listed as an alien and she was born in Virginia !!! Census data isn't always that accurate.
Agnew's father was naturalized prior to September 12,1918. This can be seen from his WW I draft card registration dated Sept 12,1918. One of the fields checked off is that he was a naturalized US citizen. Agnew was born in November, so his father was a US citizen at the time of his birth.