Average-case analysis of QuickSort and Binary Insertion Tree height using incompressibility

We study the Kolmogorov complexity of a Binary Insertion Tree, and present a succinct encoding scheme for Binary Insertion Trees produced from incompressible permutations. Based on the encoding scheme, we obtain a simple incompressibility argument that yields an asymptotic analysis of the average height of a Binary Insertion Tree. This argument further implies that the QuickSort algorithm sorts a permutation of n elements in Θ(nlgn) comparisons on average.