How Many Stemmata with Root Degree k?

We are investigating parts of the mathematical foundations of stemmatology, the science reconstructing the copying history of manuscripts. After Joseph Bédier in 1928 got suspicious about large amounts of root bifurcations he found in reconstructed stemmata, Paul Maas replied in 1937 using a mathematical argument that the proportion of root bifurcating stemmata among all possible stemmata is so large that one should not become suspicious to find them abundant. While Maas’ argument was based on one example with a tradition of three surviving manuscripts, we show in this paper that for the whole class of trees corresponding to Maasian reconstructed stemmata and likewise for the class of trees corresponding to complete historical manuscript genealogies, root bifurcations are apriori the most expectable root degree type. We do this by providing a combinatorial formula for the numbers of possible so-called Greg trees according to their root degree (Flight, 1990). Additionally, for complete historical manuscript trees (regardless of loss), which coincide mathematically with rooted labeled trees, we provide formulas for root degrees and derive the asymptotic degree distribution. We find that root bifurcations are extremely numerous in both kinds of trees. Therefore, while previously other studies have shown that root bifurcations are expectable for true stemmata, we enhance this finding to all three philologically relevant types of trees discussed in breadth until today.