Tree rearrangement graphs admit paths of decreasing Robinson-Foulds distance

TL;DR

This paper studies the behavior of Robinson-Foulds distance along paths in NNI and SPR tree rearrangement graphs, showing monotonic decrease properties and limitations in certain cases.

Contribution

It proves that any two trees can be connected by paths with decreasing RF distance in NNI and SPR graphs, revealing new insights into treespace navigation.

Findings

RF distance decreases monotonically in NNI paths

RF distance decreases strictly in SPR paths

Existence of tree pairs with no strictly decreasing NNI path

Abstract

Tree rearrangements such as Nearest Neighbor Interchange (NNI) and Subtree Prune and Regraft (SPR) are commonly used to explore phylogenetic treespace. Computing distances based on them, however, is often intractable, so the efficiently computable Robinson-Foulds (RF) distance is used in practice. We investigate how the RF distance behaves along paths in the NNI and SPR graphs, where trees are nodes, edges represent single rearrangements. We show that any two trees are connected by a path along which the RF distance to the target decreases monotonically in the NNI graph and strictly in the SPR graph; we also exhibit trees for which no strictly decreasing NNI path exists.