# New reduction rules for the tree bisection and reconnection distance

## TL;DR

This paper introduces five new reduction rules that improve the bounds on the size of reduced unrooted phylogenetic trees related to the TBR distance, enhancing previous reduction techniques.

## Contribution

The paper presents the first reduction rules that surpass existing subtree and chain reductions, further decreasing the size bound of reduced trees in phylogenetics.

## Key findings

- Reduced trees have at most 11k-9 taxa after applying new rules.
- New rules combine unrooted generator approach with agreement forest analysis.
- Bound on tree size is tightened from 15k-9 to 11k-9.

## Abstract

Recently it was shown that, if the subtree and chain reduction rules have been applied exhaustively to two unrooted phylogenetic trees, the reduced trees will have at most 15k-9 taxa where k is the TBR (Tree Bisection and Reconnection) distance between the two trees, and that this bound is tight. Here we propose five new reduction rules and show that these further reduce the bound to 11k-9. The new rules combine the ``unrooted generator'' approach introduced in [Kelk and Linz 2018] with a careful analysis of agreement forests to identify (i) situations when chains of length 3 can be further shortened without reducing the TBR distance, and (ii) situations when small subtrees can be identified whose deletion is guaranteed to reduce the TBR distance by 1. To the best of our knowledge these are the first reduction rules that strictly enhance the reductive power of the subtree and chain reduction rules.