ON THE AVERAGE INTERNAL PATH-LENGTH OF M-ARY SEARCH-TREES

Consider an m-ary tree constructed from a random permutation of size n. When all permutations are equally likely, the average internal path length, which may be considered as a cost measure for searching the tree, is shown to be (n plus 1)H//n/(H//m minus 1) plus cn plus O(n** beta ), beta less than 1, with c equals c(m) equals minus m/(m minus 1) minus (H//m minus 1)** minus **1 plus A//m**(**m**), where H//k is the k**t**h harmonic number and A//1**(**m**) is a coefficient obtained by solving a linear system of equations. This result tells us that the average cost of searching unbalanced m-ary trees is essentially the same as that of searching other popular variants of m-ary trees like B-trees and B** plus -trees where sophisticated methods are used for balancing.