# An analysis of the Geodesic Distance and other comparative metrics for tree-like structures

## TL;DR

This paper provides a comprehensive analysis of thirteen metrics for comparing tree-like structures, focusing on their discriminatory power, computational complexity, and introduces a new representation method for trees.

## Contribution

It formalizes and compares multiple tree comparison metrics, introduces a new tree representation, and analyzes the Geodesic Distance in depth with complexity insights.

## Key findings

- Thirteen metrics for tree comparison are systematically analyzed.
- A new tree representation method is proposed.
- The Geodesic Distance's worst-case complexity is discussed.

## Abstract

Graphs are interesting structures: extremely useful to depict real-life problems, extremely easy to understand given a sketch, extremely complicated to represent formally, extremely complicated to compare. Phylogeny is the study of the relations between biological entities. From it, the interest in comparing tree graphs grew more than in other fields of science. Since there is no definitive way to compare them, multiple distances were formalized over the years since the early sixties, when the first effective numerical method to compare dendrograms was described. This work consists of formalizing, completing (with original work) and give a universal notation to analyze and compare the discriminatory power and time complexity of computing the thirteen here formalized metrics. We also present a new way to represent tree graphs, reach deeper in the details of the Geodesic Distance and discuss its worst-case time complexity in a suggested implementation. Our contribution ends up as a clean, valuable resource for anyone looking for an introduction to comparative metrics for tree graphs.