Not quite and even if, on a too slow scale. While trees try hard to spread, to survive they need minimum environmental conditions and e.g. protection from plant

Not quite and even if, on a too slow scale. While trees try hard to spread, to survive they need minimum environmental conditions and e.g. protection from plant eaters while they are small. Take the extension of the Sahara to the south in the 70ies. There were droughts, trees being cut and new trees being eaten by goats. Once the soil had become unprotected, the desert would extend. Fortunately, there are now efforts to create the "great green wall" where new trees are planted and protected until they are strong enough to stand for themselves. This seems to be quite successful.