botany - Do trees really get a large share of their mass from the carbon in the air?

Yes. In fact the organic compounds' mass comes mostly from the air, since Photosynthesis essentially builds up glucose by only adding hydrogen to CO₂. The 2 H₂O → 2H₂ + O₂ reaction can be treated seperately, as was determined by Sam Ruben and Michael Kamen with ¹⁸O isotope tracing, i.e. in fact only the hydrogen in the carbohydrates comes from the soil, and this has of course a much smaller mass.

As said by Nick T, the more complex compounds also incorporate other elements that the plants get from the soil, but most of them still consist mainly of C and O. The total mass of a tree of course also has a lot of water and some minerals in it, but it's still safe to say that a tree "consists mainly of air".

To your question about surface area and air CO₂ content, neither is usually the limiting factor: in nature, light exposure, water and mineral supply are (although in enclosed spaces like greenhouses CO2 fertilization is sometimes necessary).