In the exposition (or beginning of the story that gives some background information and sets things up), O.Henry describes the section of town called Greenwich village, and describes the various artists and free-thinkers that inhabit it. Among those are Sue and Johnsy, female roommates that share a loft together, and are aspiring artists.
There are several points of complication, or rising action. The first is that O. Henry mentions that pnemonia has struck the village. The second is that Johnsy gets it, and the third is that the doctor has not much hope for her. Lastly, Johnsy declares that as soon as the last leaf falls off the ivy outside her window, she will too "go away."
This leads to the climax, where that night Sue and their unique and surly neighbor Behrman, wait the night, to see if Johnsy passes. Sue opens the window the next morning to find the leaf there.
The falling action is that the leaf stays there, and Johnsy gets better.
The resolution is when Sue tells Johnsy that the leaf wasn't real, that Behrman painted it, and subsequently died from pnemonia himself. I provided a link to the story below; it isn't very long, and it's a good one. I hope this helps!
Exposition in a story refers to the provision of essential information which provides background to the events about to unfold. It informs the reader about the setting, conditions, circumstances and some background to the characters in the story. Without such context, the story may seem meaningless.
In the exposition of "The Last Leaf," the writer informs us that Sue and Johnsy were two artists who shared a studio in an art colony in Greenwich Village and that the community was under attack by a disease, pneumonia, which during a very cold winter killed a large number of residents in the area. It also informs us that Johnsy was struck by the disease and became extremely ill.
The complication refers to the difficult circumstances that the character/s find themselves in and their attempts to find solutions. The complication is sometimes caused by a conflict, whether external or internal. The complication in our story is found in Johnsy's refusal to fight the disease, literally giving up and accepting the fact that she is going to die. The conflict arises from Sue's and the doctor's attempts to encourage her to fight back. To add to this, Johnsy has decided that she will die as soon as the last leaf has fallen from a vine that she can see through her window. Further complication lies in finding something to motivate Johnsy to want to continue living since she is so determined to die.
The action in a story builds to a climax and the falling action refers to events that occur once the climax has been reached. The climax in this story occurs when there is only one leaf left on the vine. The climactic expectation is that once this leaf has fallen, Johnsy will die. Johnsy survives, however, because when she looks out of her window she notices the leaf still clinging to the vine and this inspires her to live. The falling action, therefore, is the description of Johnsy's recovery after this dramatic discovery —one that saves her life.
The resolution in a story generally refers to a point when a conflict is resolved and the story comes to an equitable and satisfying end. In this story, the resolution is achieved when Johnsy is fully recovered and learns of Mr. Behrman's unselfish sacrifice—he had finally painted his masterpiece and saved her life, but tragically lost his own in the process.