Dead Trees are trees that have reached their maximum lifespan. The only way to find this is to wait for a tree to reach its maximum size and wait for it to turn black and/or branches falls without interaction. When the player cuts the tree, the stump will turn black a few seconds later after the tree was cut. The logs will also turn black but that takes a much longer time. When a tree is about to die, the player can identify it because it has no leaves (Trees that don't spawn with leaves, like Gold Trees, are exempt from this). Once the leaves are gone, the tree will die a couple minutes later.
The trees will turn black and, seconds later, the tree will fall over, after that there will be a pile of black logs left. After some time, the black dead logs will despawn.
The Dead Tree is commonly mistaken for Spook Trees or Sinister Trees by newer players because it almost has the same color. This wood isn't collectible and any tree can die. No matter what kind of tree it was living, dead trees will always stay the same black color.