Since decline of civilizations and deforestation are a recurring theme in history, this time will be no different. Deforestation, worldwide, has been underway for centuries, even though many societies have taken steps to protect their forests. As always, location is everything. It all comes down to population density and competition for local resources, and society's ability to utilize said resources. Trees don't harvest themselves, and most folks these days don't have the ability to utilize wood as heat. This, of course, will change to some extent.
The two large trees in the center of the photo are tulip poplars (Liriodendron tulipifera) and reproduce at an amazing rate, almost like weeds. Good for many uses.

In our case, the forest regenerates much faster than our needs could deplete them. Again, location, population, along with efficient structures and passive solar. We have much more deadfall than we need or could utilize, barring a big increase in local population and demand. Further, the several acres I've let go fallow have been growing trees since the moment the cattle were removed. In ten years, dense young forests have sprung up. Many of these trees (tulip poplars mainly) can be utilized by coppicing; my 'Plan B' if we're ever forced to cook with wood. Meanwhile, I'm heading out to our logging road to harvest two large standing dead oaks that will easily get us through the next month or more. We rarely harvest living trees due to availability of natural die-off; again, much more than we would ever utilize.