Becky,
Thanks, I like purty things...
Greg, each individual roof support, unless it is massive, is not able to support very much on its own, especially in the center of the span.
However, when you have several minimal supports a short distance apart, and all connected together with the exterior roof, and sometimes an interior ceiling, it becomes very strong and rigid.
There are span tables available to help size the supports, but they are designed for homes/buildings that must consider different types of loads, and so are not a lot of help.
The TD community, with thousands built, is a pretty good indicator of what works.
The average size seems to be between a 1x2 and a 2x2.
While size is important, it is more critical to choose a strong species without defects.
A Pine 2x2 full of knots is far weaker than a straight grained 1x2 piece of Poplar...
Spacing, or centers also is a determining factor.
Placing the supports 24" on center may leave you with a weak roof, but moving the supports to 12" centers may allow you to entertain your friends on the roof with no, or very little deflection.
Rob.
Waiting for "someday" will leave you on your deathbed wondering why you didn't just rearrange your priorities and enjoy the time you had, instead of waiting for a "better" time to come along...