Different phylogenetic methods may be better suited to rooted or unrooted trees. If
a method produces an unrooted tree, then the root (i.e., the common ancestor of all taxa)
could theoretically be placed anywhere. Thus, there will be more rooted binary trees than
unrooted binary trees on the same number of taxa. The question is: how many more rooted trees
As in the case of unrooted trees, say that we have a fixed collection of $n$taxa
labeling the leaves of a rooted binary tree $T$. You may like to verify that (by extension of “Counting Phylogenetic Ancestors”) such a tree will contain $n-1$ internal
nodes and $2n-2$ total edges. Any edge will still encode a split of taxa;
however, the two splits corresponding to the edges incident to the root of $T$ will be equal.
We still consider two trees to be equivalent if they have the same splits (which
requires that they must also share the same duplicated split to be equal).
Let $B(n)$ represent the total number of distinct rooted binary trees on $n$ labeled taxa.