From my understanding, a "controlled" burn can be done for several reasons, two of which are: eliminating the possibility of a forest fire in dry or dense areas and also a controlled burn will cause the burned out area to grow back rich and fertile.

Yup, the controlled burn does as grandmapoo said - reduces the fuel load to prevent uncontrolled fires. A fire stimulates native plant growth that is good wildlife habitat. Fire also helps control invasive shrubby growth that is not adapted to fire. Invasive shrubby growth can overtake wildlife habitat, decreasing diversity making it less friendly for critters.

They do it in very small areas - probably in order to keep it under control. I imagine the have an alternating rotation on the areas that are burned.

Invasive shrubby growth, now that sounds like what I want to do away with in my yard, lol. Don't think I'll be burning it off, tho.

If this makes for more fertile land won't the invasive plants come back full force? I'm just curious as I used to hike and horseback ride and see the things they were doing. Don't get out there as much now.....sigh.
Kiki

Actually it allow seeds to sprout, that wouldn't be able to under dense conditions.
Kiki, I actually do use some of the shrubs in my woods in my woodland garden, but keep them trimmed and manageable. :)

Invasive alien species are not adapted to fire and generally they are killed because they don't have the thick bark or deep root systems that the native plants have developed over thousands of years evolving with fire...

I sure would love to burn some areas in my woodlands, but it would be just too hard to be sure it didn't take off and destroy everything.

Yes, in order to burn a woodland (or any land really) you have to pay strict attention to fuel load and weather conditions.

Catherine, there are professionals who land owners can hire to do a burn for them. They would evaluate the property, construct the proper fire breaks and bring in a trained crew to do the burn.

A small game reserve near me burns undergrowth to stimulate the growth of native grasses for wild turkey chicks.

Jones & Foote in "Gardening with Native Wild Flowers" recommend burning for maintaining meadows for similar reasons that grandmapoo and joepyeweed have said. Meadows here in the SE would revert to forest otherwise.