UNROOTED SUPERTREES Limitations, traps, and phylogenetic patchworks

Whereas biologists might think of rooted trees as the natural, or even the only, way to display phylogenetic relationships, this is not the case for a mathematician, to whom rooted and unrooted trees are graph-theoretical constructions that can be transformed easily into one another. An unrooted tree contains the same information as its rooted counterpart with the single exception that it does not tell you where the “evolutionary process” started. Rooting a tree is often more of an art than a science, and a pressing problem in systematic biology is precisely the exact placement of a root. In addition, many phylogenetic algorithms in fact output unrooted trees that are rooted (artificially) in a subsequent step. From this, it is clear that finding an unrooted supertree or parent tree is of the same interest as it is for the rooted case. But, whereas a single unrooted tree can always be transformed into a rooted tree carrying the same information, this is no longer the case for collections of unrooted trees. Hence, the supertree problem for rooted trees is a special case of that for unrooted trees. As is often the case, this means that many things that can be done with rooted trees (the special case) are no longer valid for unrooted trees (the general case). In fact, the smallest possible example of a collection of unrooted trees that cannot be transformed into a collection of rooted trees is already sufficient to demonstrate that, unfortunately, many convenient features of the rooted supertree problem do not carry over to the unrooted supertree problem. On the positive side, if the set of input trees fulfills some minimality criterion, then there exists a simple set of conditions to check whether there is exactly one parent tree for this collection. In addition, the unique parent tree, should one exist, can be constructed quickly because the set of input trees always shows a certain “patchwork” structure.