Extremal Distances for Subtree Transfer Operations in Binary Trees

TL;DR

This paper investigates the extremal and average case complexities of subtree transfer operations in binary trees, with implications for phylogenetics, providing bounds on the number of moves needed for transformations.

Contribution

It extends existing results by establishing bounds on the maximum and expected number of subtree transfer moves between binary trees, including for multiple trees.

Findings

Maximum moves required is n - Theta(sqrt(n)).

Expected moves for random trees is n - Theta(n^{2/3}).

Results extend to multiple trees via agreement forests.

Abstract

Three standard subtree transfer operations for binary trees, used in particular for phylogenetic trees, are: tree bisection and reconnection ($TBR$), subtree prune and regraft ($SPR$) and rooted subtree prune and regraft ($rSPR$). For a pair of leaf-labelled binary trees with $n$ leaves, the maximum number of such moves required to transform one into the other is $n-\Theta(\sqrt{n})$, extending a result of Ding, Grunewald and Humphries. We show that if the pair is chosen uniformly at random, then the expected number of moves required to transfer one into the other is $n-\Theta(n^{2/3})$. These results may be phrased in terms of agreement forests: we also give extensions for more than two binary trees.