Note on islands in path-length sequences of binary trees

TL;DR

This paper revisits the characterization of path-length sequences in binary trees, reformulating it through an integrality condition on a scaled Kraft sum related to the tree's structure.

Contribution

It introduces a new formulation of the topological ordering condition using an integrality criterion on the scaled Kraft sum of subsequences.

Findings

Reformulation of the characterization using integrality conditions

Connection between scaled Kraft sum and ancestor counts at specific levels

Enhanced understanding of binary tree path-length sequences

Abstract

An earlier characterization of topologically ordered (lexicographic) path-length sequences of binary trees is reformulated in terms of an integrality condition on a scaled Kraft sum of certain subsequences (full segments, or islands). The scaled Kraft sum is seen to count the set of ancestors at a certain level of a set of topologically consecutive leaves is a binary tree.