According to Aristotle's definition, what are the elements of tragedy found in Macbeth?

In his Poetics, Aristotle defines "tragedy"
in part by stating that "the writer presents 'incidents arousing pity and fear, wherewith to
interpet its catharsis of such emotions.'"  Under this definition, Macbeth
functions as a tragedy because the events in the play do arouse the pity and fear of the
audience, yet by the end of the play, the audience should no longer relate to these emotions.  At
the beginning of Macbeth, the witches predict that Macbeth will become the
Thane of Cawdor and the King of Scotland.  Although Macbeth sees great happiness in this
prediction, Banquo fears the absolute verity of the witches' prediction and says that caution
must be taken lest they fall into a trick.  The witches represent supernatural elements and
should arouse fear in the audience.  When the prediction begins to come true, Macbeth cannot rid
himself of his greedy, ambitious desires and the audience pities Macbeth for not having a
stronger, more cautious character.  He is a valiant soldier who has been led astray.  However,
these feelings of pity give way at the end of the play when Macbeth himself admits that he has
behaved in the wrong manner yet still goes on to fight Macduff just to see it through to the
end.  Macbeth never makes amends for his wrongs, and this is tragic according to Aristotle's
definition.