Madeline only appears three times in "The Fall of the House of Usher." How do her appearances develop the plot of the narrative?By Edgar Allen Poe.
First, in order to answer this question, it needs to be determined what the conflict of "The Fall of the House of Usher" is because conflict defines plot since plot is the working out of the main conflict, the complications and the resolution of the conflict. And there is more than one view on what the conflict of this short story by Edgar Allen Poe is.
One view holds that the conflict is (1) rationality versus irrationality, or insanity. In this view, the narrator is rational while Usher is irrational, or insane. The view has difficulties because it cannot explain how everything seemingly "irrational" that was perceived by both the narrator and Usher was proven in the resolution to be actual fact, i.e., Madeline did walk through the door in her funeral shroud; the House of Usher did crumble and disintegrate into the tarn, thus ending along with the family of Usher; the vegetation did prevail.
Another view holds that the conflict is imagination, as defined by the Romantics, versus expected perception. This view of the conflict is supported in the early portion of the story where the narrator talks about "had so worked upon my imagination" and "what must have been a dream." This view is a strong one but doesn't adequately encompass the question of the debate regarding the sentience of vegetation.
A third view holds that the conflict is the appearance of imagination, again according to the Romantic era definition, versus the presence of life, or sentience. This view has a scope that can encompass the other two as follows. Is it rational or irrational to believe in the sentience and intentionality of vegetation? Can imagination (the higher order of reason that creates metaphorical meaning for and order in the world) be trusted to correctly order elements of the world even when perception violates expectation as the narrator experienced upon going to the House of Usher?
If the third view is employed in a discussion of Madeline, then it can be said that Madeline drives and complicates the plot (derived from conflict) of the story by illustrating the conflict of the appearance of imagination (reason) versus the presence of life, or sentience. At the narrator's first glance of her, Madeline was semi-catatonic. She was later fully catatonic and unconscious and lacking any display of imagination, yet she was alive, or sentient as was later proved by her ghastly appearance at the chamber door. Therefore Madeline parallels the vegetable domain, which is similarly "catatonic," "unconscious," and lacking a display of imagination and yet is -- according to Usher and the experience of the House of Usher (both building and people) -- alive, or sentient.