PLOT

As for the plot, It consists of exposition, rising action, climax, falling action and resolution. In the exposition 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.

There are several points of  rising action.  The first is that O. Henry mentions that pneumonia 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 pneumonia himself.

The types of speech employed by the author of the analysed story are narration and dialogues.