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B Trees

Asymptotics for tree height

Which statement gives you more information about a hotel?

A) The most expensive room in the hotel is $639 per night.
B) Every room in the hotel is less than or equal to $639 per night.

The answer is A, since we know for sure that there is atleast one room in the hotel that costs $639, whereas there is no guarantee of that in B. Moreover, a large number of hotels, (or motels) would fulfill the statement in B (ie, have rooms that cost a 100 bucks, but would still fulfill the statement).

In the same vein, the one of the following statements about tree height is more informative:

A) Worst case BST height is Θ(N).
B) BST height is O(N).

In this case, it's A.

Which leads to the question, why is Big O useful? It's because of the following:

  • Allows us to make simple blanket statements, e.g. can just say “binary search is O(log N)” instead of “binary search is Θ(log N) in the worst case”.
  • Sometimes don’t know the exact runtime, so use O to give an upper bound.