I'd go with @Aarthi's load bearing table resource for a general idea of what's reasonable.
If you're looking for equations though, you can start with these:
Beam Deflection Formulas
Beam Deflection and Stress Calculator
Area Moments of Inertia
Using the Parallel Axis Theorem
Wood Material Properties (Modulus of Elasticity (E) found in Table 4-3a)
For the dynamic loading, you'll want to do something similar to the fun I had on this question.
...and you may want to consult a good Mechanics of Materials book. (cheaper paperback international edition on Ebay)
As @Ian points out, the problem is not a simple one and is best solved by simply using what's worked for other people in the past. Go take a look at the swings at your local park and use the same size of beam, provided the span is comparable.
Also, if you're really worried, you could always make the rope into a 'Y' to eliminate bending stress on the beam, leaving it solely in shear. This way, the beam is bearing the compression load from the lateral tension on the 'Y', which will keep the trees from bowing toward each other.
Diagram:
| |______________________| |
| | | | | |
| tree | | | | tree |
| |__|_________________|_| |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | \ / | |
| | Y | |
| | | | |
| | | | |
| | | | |
...more rope and trees...
| | | | |
| | | | |
| | ----- | |
| | / ___ \ | |
| | | / \ | | |
| | | \___/ | | |
| | \ / | |
| | ----- | |
Here is the answer to my actual question:
http://awc.org/codes-standards/calculators-software/spancalc
Edit: Sorry if I'm not on the same wavelengths as anyone else. I already know about insurance and inspectors, and that was not at all what I was asking about. I just wanted to know what length of wood would span a certain distance given various criteria. I would have expected an answer to that question, instead of all the "your a dumb homeowner who's going to get in trouble" type replies. I question the idea that there will be 50psf of weight from snow given the fact that it is an almost non-solid top, and saw no real attempt to answer my question, so I had to come up with this. It took me awhile to find, and I hope it helps others.
Best Answer
Putting those numbers into the sagulator, a 12' 2x12" Ponderosa pine board lying flat could support 100-125 lbs with about 0.2" of sag in the middle.
If it were on edge, it could support several thousand pounds. The sagulator says borderline sag began at about 12,500 pounds, but that was beyond their stated target sag of 0.02" / ft. I found that target sag at about 7000 lbs.