More on outgroup rooting: whither root branch length?

Continuing on the theme of rooting, there is a somewhat peculiar aspect to outgroup rooting. In the previous post, when I explained five different ways of rooting a phylogenetic tree, I used the following tree to illustrate outgroup rooting:

Species A is the outgroup to the ingroup BCDE. As we can see, the long red branch separating outgroup and ingroup has been assigned entirely to the stem of the ingroup, leaving the outgroup to sit on a zero length branch. There is, however, no logical reason why this has to be so. We could just as well divide the length of that branch evenly among ingroup and outgroup:

This is what the tree viewer software FigTree appears to do automatically when asked to outgroup-root an unrooted tree.

At the other extreme, we could also assign the entire length of the branch to the outgroup. This is what the tree viewer software TreeView does as a default:

It is important to keep in mind two things. First, although at first sight the above tree looks as if it has an unresolved polytomy at the base, where I placed the blue circle, the tree is still exactly the same as in the two previous images.

The confusion that some observers may experience when faced with such a tree arises from the unconscious assumption that the connection to the rest of the tree of life also goes through the blue circle, giving that node four connections. Seeing a lot of tree figures conditions us to expect that connection to go off from the vertical line. In reality, the rest of the tree of life connects somewhere along the red internode, and if we were to make the connection the red internode would be transformed into two horizontal lines and one vertical line. Consequently, the node in the blue circle actually has only three connections, the same number as every other fully resolved tree node: one from the ancestral lineage, two towards the descendent lineages.

The second thing to keep in mind is that the assignment of the length of the red branch is, under outgroup rooting and in the absence of additional information provided by a yet more distant outgroup, totally arbitrary. So no matter which of the above three ways of doing it you will find in a phylogenetic publication, they are all fully equivalent, even as the appearance of the trees is superficially different.