When trees break down, the black wood remains and can not be interacted with, meaning it serves no purpose at all.
Trees are nearly dead when they lose their leaves with the exception of Gold Wood, Sinister Wood, and Spook Wood tree, as they are intentionally designed to appear with no leaves. During the End Times Update, all trees lost their leaves, but none of the trees were really dead, as this feature was intentional for the update.
When a tree dies, breaks down, and accidentally falls onto a vehicle, it may glitch the vehicle that the tree landed to. It also has a rare chance to be chopped, but again, it is cannot be interacted with, even when chopped.