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Most of the rolls concerning the Cherokee were produced from the 1830s through the early 1900s, so you will need the names of ancestors who were alive and members of the tribe at that time. Most people start by checking the enrollments from the period 1900 to 1910, known as the Dawes Rolls for the Oklahoma Cherokee and the Guion Miller Rolls for the Eastern Cherokee. Online name indexes are available for both of these sources, and you can get access to the full rolls through Ancestry.com and Fold3.com. These are paid genealogical websites, but they can be accessed for free at any National Archives research facility. (You may also be able to access these sites through a local library.)
The National Archives’ page on Researching an Individual or Family also has many resources for tracing Native American ancestors that you may find useful.
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