We learned in school about a Greek or Roman guy who proved through experiment that the air supplied the bulk of plant growth. He weighed the soil before and after a tree was grown in a container. Only water was added during his experiment.
Most of the nutrient value in trees is in the bark, leaves and small branches. So when lumber is milled and the waste returned to forest, the forest loses very little nutrient. Ashes from the wood stove contain some of this mineral content.
On other threads concerning rocket stove efficiency examples of ash volume and weight have been kicked around as evidence of something. Some of the stoves were in wet coastal areas and others where in dry areas with alkaline soil. It's highly likely that variations in ash volume have more to do with soil minerals than with any differences between these stoves.