In the short story 'The Last Leaf' by O. Henry, Johnsy's analogy of death of the vine happened to Mr. Behrman. Johnsy was suffering from depression due to her severe illness and had developed the belief that when the last leaf of the ivy vine outside her window would fall she would die. However, when the dreaded moment arrived, Mr. Behrman died instead of her.
Yes, it is really connected to how people endure life and face the most difficult challenge of one’s surpassing death. The story stresses that the process of recovery from a severe illness has a strong psychological aspect to it. A person's attitude towards the illness and willingness to survive affects the way medicines work. Many scientific studies have shown that optimistic people tend to recover better than pessimistic people.