Does "Arctic Greening" mean an increase in the spatial extent of vegetation, an increase in particular types of vegetation, or both?
You have an unclear question. See my comment on the question itself.

Answering just the question:  

I've not seen a formal  definition of arctic greening.  Conceptually it's replacing a landscape that is in effect a cold desert with one that has:


Longer growing season.
Greater biodiversity
Greater gross photosynthesis.


You would think that it would also be greater net photosynthesis too, but much of the arctic has large deposits of peat moss, forest duff and sedges. With warmer temperatures we will likely see greater bacterial action on the peat. Not clear which way the carbon/acre figure will move.

It's also not clear what the effect of an open oceans in the high arctic will be.  Depending on the nutrient levels in those waters could see a lot more phytoplankton. BVOC exchange between ocean and atmosphere?

You can get a first level approximation substituting latitude for time. I expect BVOCs to disperse fairly quickly so you would need to measure at ground level. Analyzing a bunch of sites that are representative of the various major communities -- Boreal forest, tree line, tundra/puckey-brush (scrub willow, birch) Tundra/moss, forbs; Rock.

For a preliminary analysis you don't even need to go to the arctic. Use a convenient mountain.

That can nail down quantities of BVOCs. Correlate that with HCOH to see how good a proxy HCOH is for BVOC.

This leaves a bunch of unanswered questions:

Is the time frame for the degradation of HCOH similar to for other BVOCs?  You may have to do radioactive tracer tests in containers set at similar conditions of temperature, pressure and UV illumination.

What are the degradation products?  What are the effects of these?  Do any of the original products or their degradation products acts as catalysts?

I think you have bitten off a big research project.