I unfortunately can't find a reference, but I heard that it takes about 20 years for a newly planted tree to peak with regards to absorption of carbon, and after about 100 years it reaches equilibrium where the forest emits about the same amount of carbon to the atmosphere as it absorbs (through decaying matter, fallen trees, and whatnot). It of course depends on the type of tree.
I think it's best to look at this as mass preservation problem. Trees absorb CO2 from air and store in their bodies. Once they die, the CO2 gets released.
Since there's a limited tree mass per area unit that the ground can support, once you reach that mass the input and output need to zero out, since there's nowhere to store any additional CO2.
I'm not an expert in this area, but from what I can tell:
Young rapidly growing forests sequester carbon more quickly than mature forests, but mature forests still sequester some carbon, possibly into the soil. (One source https://www.sciencedaily.com/releases/2008/09/080910133934.h... )
Separately, the regrown forest could be maintained in a state of peak carbon sequestration by doing selective logging from time to time, and as a bonus it would be profitable to do so.
