Proof that the height of a balanced binary-search tree is log(n)

Question

The binary-search algorithm takes log(n) time, because of the fact that the height of the tree (with n nodes) would be log(n).

How would you prove this?

Answer 1

Just look up the rigorous proof in Knuth, Volume 3 - Searching and Sorting Algorithms ... He does it far more rigorously than anyone else I can think of.

http://en.wikipedia.org/wiki/The_Art_of_Computer_Programming

You can find it in any good Computer Science library and on the bookshelves of many (very) old geeks.