Using the concept of a "tree-growing" search strategy, we prove that for most practical insertion sorting algorithms, the number of comparisons needed to sort n keys has asymptotically normal behavior. We prove and apply a sufficient condition for asymptotically normal behavior. The condition specifies a relationship between the variance of the number of comparisons and the rate of growth in height of the sequence of trees that the search strategy "grows."
"On tree-growing search strategies." Ann. Appl. Probab. 6 (4) 1284 - 1302, November 1996. https://doi.org/10.1214/aoap/1035463333