Should Static Search Trees Ever Be Unbalanced?

In this paper we study the question of whether or not a static search tree should ever be unbalanced. We present several methods to restructure an unbalanced k-ary search tree T into a new tree R that preserves many of the properties of T while having a height of log k n+1 which is one unit off of the optimal height. More specifically, we show that it is possible to ensure that the depth of the elements in R is no more than their depth in T plus at most log k log k n+2. At the same time it is possible to guarantee that the average access time P (R) in tree R is no more than the average access time P (T ) in tree T plus O(log k P (T )). This suggests that for most applications, a balanced tree is always a better option than an unbalanced one since the balanced tree has similar average access time and much better worst case access time.