I have read a news story about how a Native American tribe has been able to buy back an island from private ownership in their historic lands.

It is my understanding that some (all?) Native American tribes have a degree of sovereignty and control over some lands – for instance being able to build casinos on their lands where the laws of a state would not permit it.

Would any land acquired by the tribe automatically become a part of these (sovereign) tribal lands, or would it remain subject to the laws of the state? If not automatically, is there a process for the land to become a part of their sovereign soil?

By analogy here: if the Canadian government were to buy a random property in Berlin, I would think that that it would remain a part of Germany, not become a part of Canada. Whereas there have been times in history in which a government has purchased both land and sovereignty over it – e.g. the Louisiana purchase, which I'd imagine required an international treaty.

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