The path length of binary trees

Abstract

More than twenty years ago Nievergelt and Wong obtained a number of new bounds on the path length of binary trees in both the weighted and unweighted cases.

For the unweighted case, the novelty of their approach was that the bounds were applicable to all trees, not just the extremal ones. To obtain these “adaptive” bounds they introduced what came to be known as the weight balance of a tree, subsequently used as the basis of weight-balanced trees.

We introduce the notion of the thickness, Δ(T), of a tree T; the difference in the lengths of the longest and shortest root-to-leaf paths in T. We then prove that an upper bound on the external path length of a binary tree is

$$N(\log _2 N + \Delta - \log _2 \Delta - 0.6623),$$

where N is the number of external nodes in the tree. We prove that this bound is tight up to an O(N) term if Δ ≤ $\sqrt N$. Otherwise, we construct binary trees whose external path length is at least as large as N(log₂ N + φ(N, Δ) Δ − log₂Δ − 4), where φ(N, Δ) = 1/(1 + 2 Δ/N).