Does natural plant decomposition release more greenhouse gases than burning
Another way of looking at the question:

I take 2 standard wheelbarrows of bits of plants from my garden. One wheelbarrow is used to fill the compost bin. The other goes to the bonfire.
How much ash do I have from burning compared to how much compost from the compost bin. Any gardener will know that the amount of compost will be much larger than the amount of ash. So now we need to compare the chemical composition of the ash with that of the compost. For Wood Ash there are quite a few research articles. Most of the carbon atoms in wood ash will be in the form of Calcium Carbonate (CaCO3). Percentage in the ash will vary widely but lets take a median of 50%. Ash as a percentage of original organic matter will be maybe of the order of 8% by weight. 

For an original 100kg of organic matter (yes, its a big wheelbarrow) we would have 8kg of ash. In this ash would be 4kg of CaCO3. The carbon atoms would be 12% of that (relative atomic masses). So 100kg of organic bits gives you 0.48kg of carbon atoms left in the ash when you burn it.

There seems to be fewer research articles on composting but we found the following approximations. For 100kg of organic matter there will be about 9% organic carbon  (9kg). Composting will turn 100kg of raw material into around 30kg of compost in which the organic carbon content will be around 25% ( 7.5kg of carbon). These figures are median "pot luck selection" as results can vary widely. We can however see that composting retains massively more (7.5kg) of carbon atoms compared with burning (0.48kg). The difference will be in the atmosphere.