"How big around" sounds like the circumference to me. Do you mean to make primitive arrows? Arrow shafts were made out of shoots, such as dogwood, wild rose, ash, birch, chokecherry, and black locust. Reeds from common reed grass were also used with some frequency throughout North America with the exception of the Plains where
reeds did not grow. Shoots were shaved, sanded, or heat and pressure straightened. Tools made of bone or sandstone were used to straighten the shaft wood. Because they are hollow and light, reed-shaft arrows typically have a wooden foreshaft and sometimes a wooden plug for the nock end of the arrow. If a foreshaft was used, it could be glued to the main shaft, tied with sinew, or fit closely enough to not need glue or sinew.
That seems like a lot of work, so I would probably take George's advice. Even Ishi was impressed with early 20th century glue, and he would have been first in line for a Bitz, fletching tape,and mass produced shafts.