Actually, that's a specific thing that does not work (at least long term). Trees die and rot if not cut down, and the carbon from that process has to be absorbe

Actually, that's a specific thing that does not work (at least long term). Trees die and rot if not cut down, and the carbon from that process has to be absorbed by the younger trees growing in the same space. So once a (mature) forest exists at a given location, it will no longer absorb any more carbon unless some of the trees are cut down and used for something that doesn't allow their carbon content to return to the atmosphere.