Just to clarify the plant anatomy, the petiole is the botanical term for what most folks call a leaf stem. The leaf stem is what connects the leaf itself to the main vertical growing stem that is rooted in the soil.
When a leaf dies, so does its petiole or leaf stem, so it is best aesthetically to remove both the leaf and the leaf stem as soon as they start to discolor. Cut it right where the leaf stem attaches to the main cervical stem. There is no harm in leaving the dying leaf in place, except that it is not very pretty!
