  The concept of $k$-compatibility measures how many phylogenetic trees it
would take to display all splits in a given set. A set of trees that display
every single possible split is termed a \textit{universal tree set}. In this
note, we find $A(n)$, the minimal size of a universal tree set for $n$ taxa. By
normalising the $k$-compatibility using $A(n)$, one can then compare
incompatibility of split systems across different taxa sizes. We demonstrate
this application by comparing two SplitsTree networks of different sizes
derived from archaeal genomes.
