The Shinnecock tribe has an interesting argument in favor of their federal recognition. From the East Hampton Star:

“The Shinnecocks have since sued the Department of Interior in federal district court to be placed on the list. They maintain it is illegal to not place them on the list after they have received federal judicial recognition.

“The tribe also filed an amended complaint incorporating documents that show that the tribe was listed in a 1914 Department of Interior report to the United States Congress as a tribe in New York State subject to federal jurisdiction with federally protected lands. This was reaffirmed in other department lists dated 1929, 1938, and 1941, according to the Shinnecocks.

“Mr. Gumbs noted that the tribe was also included in a book “The Handbook of Federal Indian Law,” compiled by Felix Cohen for the Department of the Interior in 1945, which includes the Shinnecocks in its listing of tribes in New York State. The tribes listed here and in the other department lists have all since been added to the department’s current list of federally recognized tribes or have successfully sued to be placed on it without having to go through the Bureau of Indian Affairs review, he said.”

More details of the tribe’s claims are here.