The estimated cost of a search tree on binary words

The problem of constructing a binary search tree for a set of binary words has wide applications in computer science, biology, mineralogy, etc. Shannon considered a similar statement in his optimal coding theorem. It is NP-complete to construct a tree of minimum cost; therefore, the problem arises of finding simple algorithms for constructing nearly optimal trees. We show that there is a simple algorithm for constructing search trees sufficiently close to the optimal tree on average. By means of this algorithm we prove that for the optimal tree the average number of bits to be checked is near to its natural lower bound, i.e., the binary logarithm of the number of given words: their difference is less than 1.04.