Characterization of Trees with Maximum Security

TL;DR

This paper analyzes the security measure in binary trees, classifies tree families with maximum security, and discusses extremal properties related to maximum vertex rank.

Contribution

It provides a classification of binary tree families that maximize the sum of vertex ranks and explores extremal rank properties.

Findings

Identifies tree families with maximum security

Classifies structures maximizing sum of ranks

Discusses extremal maximum rank results

Abstract

The rank (also known as protection number or leaf-height) of a vertex in a rooted tree is the minimum distance between the vertex and any of its leaf descendants. We consider the sum of ranks over all vertices (known as the security) in binary trees, and produce a classification of families of binary trees for which the security is maximized. In addition, extremal results relating to maximum rank among all vertices in families of trees is discussed.