The trees will turn black and, seconds later, the tree will fall over, after that there will be a black pile of logs left. After some time, the black dead logs will despawn.
The Dead Tree is commonly mistaken for Spook Wood or Sinister Wood by newer players because it almost has the same color. This wood isn't collectible and any tree can die. The dead tree will look the same on every tree. The only exception is Cavecrawler Wood because when it dies, the logs will stay neon but they will become green/grey. Palm Wood keeps its slate texture when it dies. The same thing happens to Phantom Wood which keeps its sand texture when it dies.