Search times for binary search tree

Question

Does anyone know how to figure out search time for a binary search tree(i.e. worst-case, best-case, and average-case)?

Answer 1

Might want to tag this one as "homework". Here's a good starting point: http://en.wikipedia.org/wiki/Binary_search_tree

In general, a balanced binary search tree has a worst-case lookup of O(log n), best case of O(1) (when the desired value is the root) and an average case of O(log n) (the leaves contain exponentially more values than their parents).

The worst case is the most interesting and is easily seen by recognizing that the first level of a binary tree has 1 node, the second has 2, the third has 4 and so on. Thus, the number of nodes in a binary tree of depth n is precisely 2^n - 1. The mathematical inverse of the exponential function is the logarithm, thus: O(log n).

An unbalanced tree can be as bad as a linked list and may have a shape like the following:

  1
 / \
    2
   / \
      3
     / \
        4
       / \

In this situation, the worst-case access time is O(n).