# Ancestral sequence reconstruction with Maximum Parsimony

## TL;DR

This paper investigates the conditions under which Maximum Parsimony reliably infers ancestral states in phylogenetics, proving a conjecture for even character states and exploring limitations for odd states.

## Contribution

It proves a conjecture regarding the number of species needed for unambiguous ancestral state inference using MP for even character states, and discusses cases for odd states.

## Key findings

- Proved the conjecture for even number of character states.
- Identified limitations of MP for odd number of character states.
- Provided positive results for certain cases with odd states.

## Abstract

One of the main aims in phylogenetics is the estimation of ancestral sequences based on present-day data like, for instance, DNA alignments. One way to estimate the data of the last common ancestor of a given set of species is to first reconstruct a phylogenetic tree with some tree inference method and then to use some method of ancestral state inference based on that tree. One of the best-known methods both for tree inference as well as for ancestral sequence inference is Maximum Parsimony (MP). In this manuscript, we focus on this method and on ancestral state inference for fully bifurcating trees. In particular, we investigate a conjecture published by Charleston and Steel in 1995 concerning the number of species which need to have a particular state, say $a$, at a particular site in order for MP to unambiguously return $a$ as an estimate for the state of the last common ancestor. We prove the conjecture for all even numbers of character states, which is the most relevant case in biology. We also show that the conjecture does not hold in general for odd numbers of character states, but also present some positive results for this case.