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Why you'd build something intentionally with only a 30 pound rating is beyond me. If you build the roof steep enough with steel roof panels, then it would shed snow on its own. But otherwise? What are you going to do if you are away sometime and get an unusal snow load built up on that roof? 80 lb. load is the minimum requirement in my area of New York now - except for farm buildings which can be exempted. As to your question - first it depends on the type of wood. White Pine is weak as compared to Yellow Pine and I have no idea what pine you are citing. I wouldn't use white pine for any structural load-bearing work. It's a waste of time and money. With Yellow Pine, Fir, etc. (high strength wood) you can span 15.5 feet with 2' centers with 2 X 10s and get a 30 lb. min. rating. 2 x 12s will give you a 40 lb. rating. With the headers that will span the poles and support the rafters: If the poles are spaced 8' - double 2X6s will work. If the poles are spaced 10' apart - double 2X8s are needed. If the poles are spaced 12' apart - double 2X10s are needed. If the poles are spaced 14' apart, double 2X14s or triple 3x10s are needed. For poles, I'd only consider 6X6s. Be careful when buying since many pressure-treated 6x6s are NOT rated for burial in ground if bought under 10' or 12' long. And, in my area, NO 4X4 pressure-treated poles are rated for in-ground use anymore.
It all changed when CCA got phased out, and the more expensive ACQ treatment became the standard.
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