A Best Possible Bound for the Weighted Path Length of Binary Search Trees

The weighted path length of optimum binary search trees is bounded above by Y'./3i + 2 a. + H where H is the entropy of the frequency distribution, /3i is the total weight of the internal nodes, and aj is the total weight of the leaves. This bound is best possible. A linear time algorithm for constructing nearly optimal trees is described. One of the popular methods for retrieving information by its "name" is to store the names in a binary tree. We are given n names B1, Be, , Bn and 2n + 1 frequencies 1," ", fin, aO," ", an with /3i + Y aj 1. Here ji is the frequency of encountering name Bi, and aj is the frequency of encountering a name which lies between B and B/I, a0 and an have obvious interpretations [4]. A binary search tree T for the names B1, B2, , Bn is a tree with n interior nodes (nodes having two sons), which we denote by circles, and n + 1 leaves, which we denote by squares. The interior nodes are labeled with the B in increasing order from left to right and the leaves are labeled with the intervals (Bi, B//I) in increasing order from left to right. Let b be the distance of interior node B from the root and let aj be the distance of leaf (Bi, Bi/I) from the root. To retrieve a name X, bi + 1 comparisons are needed if X B and ai comparisons are required if Bi < X < Bj/a. Therefore we define the weighted path length of tree T as: P= i bi .-I-1) + c,a,. i=1 ]=0 It is equal to the expected number of comparisons needed to retrieve a name. In [4] D. E. Knuth gives an algorithm for constructing an optimum binary search tree, i.e., a tree with minimal weighted path length. His algorithm operates in O(n 2) units of time and O(n2) units of space. In [6] we discuss the following "rule of thumb" for constructing nearly optimal binary search trees: choose the root so as to equalize the total weight of the left and right subtree as much as possible, then proceed recursively. The weighted path length of a tree constructed according to this rule is bounded above by 2+1.44.H, where H= Y fli log (1/fli)+ ce log (1/at) is the entropy of the frequency …