The coconut palm is a member of the Family Arecaceae... The old leaves break away cleanly leaving the trunk smooth.

Can you say smooth to describe a trunk? It's subjective but I would never consider the trunk of a coconut tree smooth.

In this context, it is the point at which the leaf breaks away that is left smooth - there is no residue of the leaf stalk on the surface of the trunk at that point. It is not describing the entire trunk which, as you say, not not smooth in a coconut palm.
It makes perfect sense now. Thanks Feebs11.