When trees decay, a certain portion of the material remains solid and eventually becomes part of the soil layer. Also when trees burn, a certain amount stays as

When trees decay, a certain portion of the material remains solid and eventually becomes part of the soil layer. Also when trees burn, a certain amount stays as carbon/charcoal/ash in the ground. So while some of the carbon is released when trees decay or burn, not all of it is. So I would expect a mature forest to be a carbon sink, though not necessarily a huge one.