I checked my NC Gazetteer and no Hang Dog Tree is listed, but Hanging Dog is a community in northwest Cherokee Co., NC. There is also a Hanging Dog Creek which rises in north Cherokee Co on Hanging Dog Mountain and flows southwest into Hiwassee River. It was named because an Indian's hunting dog became hung in a mass of jammed logs and vines in the flooded creek. Hanging Dog Gap is in north Cherokee in the north end of Hanging Dog Mountain. Hanging Dog Mountain extends from central Cherokee County northeast into the Snowbird Mountains in southwest Graham County.
Cherokee County was not formed until 1839 from Macon County, NC. Macon was formed in 1828 from Haywood Co., NC.
Haywood was formed in 1808 from Buncombe; so if your George Brannon was in the area in 1800, he was actually in Buncombe County.
I checked my census indexes and found George Brannon in Rutherford Co., TN in 1810 census. The only other year I found a George Brannon was in Jefferson Co., TN in 1850 and Muscogee Co., Georgia in 1850. Hope this helps. Naomi