The subtree prune and regraft distance (d SPR ) between phylogenetic trees is important both as a general means of comparing phylogenetic tree topologies as well as a measure of lateral gene transfer (LGT). Although there has been exten- sive study on the computation of d SPR and similar metrics between rooted trees, much less is known about SPR distances for unrooted trees, which often arise in practice when the root is unresolved. We show that unrooted SPR distance compu- tation is NP-Hard and verify which techniques from related work can and cannot be applied. We then present an effi cient heuristic algorithm for this problem and benchmark it on a variety of synthetic datasets. Our algorithm computes the exact SPR distance between unrooted tree, and the heuristic element is only with respect to the algorithm's computation time. Our method is a heuristic version of a fi xed parameter tractability (FPT) approach and our experiments indicate that the running time behaves similar to FPT algorithms. For real data sets, our algorithm was able to quickly compute d SPR for the majority of trees that were part of a study of LGT in 144 prokaryotic genomes. Our analysis of its performance, especially with respect to searching and reduction rules, is applicable to computing many related distance measures. 1. Introduction trees are used to describe evolutionary relationships. DNA or protein sequences are associated with the leaves of the tree and the internal nodes correspond to speciation or gene duplication events. In order to model ancestor-descendant relationships on the tree, a direction must be associated with its edges by assigning a root. Often, insuffi cient information exists to determine the root and the tree is left unrooted. Unrooted trees still provide a notion of evolutionary relationship between organ- isms even if the direction of descent remains unknown. The phylogenetic tree representation has recently come under scrutiny with critics claiming that it is too simple to properly model microbial evolution, particularly in the presence of lateral gene trans- fer (LGT) events (Doolittle, 1999). A LGT is the transfer of genetic material between species by means other than inheritance and thus cannot be represented in a tree as it would create a cycle. The preva- lence of LGT events in microbial evolution can, however, still be studied using phylogenetic trees. Given a pair of trees describing the same sets of species, each constructed using different sets of genes, a LGT event corresponds to a displacement of a common subtree, referred to as a SPR operation. The SPR distance is the minimum number of SPR operations, denoted by d SPR , that explain the topological differences between a pair of trees. It is equivalent to the number of transfers in the most parsimonious LGT scenario (Beiko and Hamilton, 2006). In general, d SPR can be used as a measure of the topo- logical difference between two trees, e.g. for comparing the outputs of different tree construction algorithms. Tree bisection and reconnection (TBR) is a generalization of SPR that allows the pruned subtree to be rerooted before being regrafted. Computation of the TBR distance (d TBR ) was shown to be NP-hard (nondeterministic polynomial-time hard) by Allen and Steel (2001), who also provided two rules that reduce two input trees to a size that is a linear functions of d TBR without altering their distance. These rules, which reduce common chains and subtrees, also form the basis of algorithms that compute the SPR distance between rooted trees (d rSPR ) (Bordewich and Semple, 2004) as well as hybridization number (h) (Bordewich et al. 2007), see Section 3.3. Such algorithms proceed as follows. First the distance problem is shown to be equivalent to counting components of a maximum agreement forest