Airplane Problem

Consider the following problem:
  1. Order the passengers by the order they get on the plane.
  2. Person 1 gets on the plane and takes a random seat.
  3. Person 2 gets on the plane and takes his/her assigned seat if it is available, or takes a random seat if not.
  4. Person 3, ..., N-1 does as Person 2 does.
  5. Person N takes the last seat remaning.

What is the probability that Person N gets his/her assigned seat?

Solution, with proof and fancy tree diagrams

Thanks to Scott Davis for posing this to me.