On the expected height of fringe-balanced trees

We study the effect of a well-known balancing heuristic on the expected height of a random binary search tree. After insertion of an element, if any node on the insertion path has a subtree of size precisely 2t+1 for a fixed integert, then the subtree rooted at that node is destroyed and replaced by a new subtree in which the median of the 2t+1 elements is the new root. IfH_n denotes the height of the resulting random tree, we show thatH_n/logn c(t) in probability for some functionc(t). In particular,c(0)=4.31107... (the ordinary binary search tree),c(1)=3.192570 ...,c(3)=2.555539 ...,c(10)=2.049289 ... andc(100)=1.623695 ....Research of the author was sponsored by NSERC Grant A3456 and by FCAR Grant 90-ER-0291