computing the quartet distance between general trees

7.1. THE ALGORITHM 43choice of indices.jnlimkFigure 7.6: Illustrates the choice of entries in Eq. (7.7).Once again, each possible pair of directed edges needs processing. And once again,since a regular undirected butterfly quartet is subject of two claims, each quartet is encounteredboth when dealing with e and when dealing with ē. Again, ē is some oppositelydirected edge, namely the one that claims the same quartet as e.The total result of counting different butterflies should be divided by four, as opposedto the shared butterflies. While similarity is identified by observing the same two leavesbehind the edge in both T and T ′ , difference is identified by requiring that the pairsof leaves behind the edges should be different. The consequence is, that if the edge pair(e,e ′ ) is identifying a different butterfly quartet, the pair (e,ē ′ ) does also and so does (ē,e ′ )and (ē,ē ′ ). Thus, one single quartet is counted as different four times and consequentlythe result should be divided by four.Contribution 7.2 It is evident that the use of directed edge claiming introduces the needfor dividing the total count by some factor. However, the article states that different butterfliesare counted twice, just like shared butterflies, and that the result should thereforebe divided by two.After implementing the diff B (T,T ′ ) algorithm I observed that the result was, “forsome reason”, consistently double of what I expected. After thorough verification thatmy implementation was correct, this led to further investigations of the correctness of thearticle. I realized that the reasoning should be different and that the result should bedivided by four, as explained in the text.