From my layman understanding: usually it's just dry undergrowth and dead stuff that burns. Getting a healthy tree to burn needs a lot of heat or time, so if you burn frequently enough the undergrowth burns away long before the trees have a chance to catch fire. If too much flammable material has built up, remove undergrowth from around the tree trunks (or inversely, stack dead branches under trees you want to burn, though not sure if anyone does that).
