The phylogenetic tree referenced does seem to work as a inf-semilattice (or sup-semilattice if you want to move from present species “upwards” to common ancestors… this depends on the direction in time you want to view things from) in that every two species have a unique common ancestor.

However, the ancestor relation in general doesn’t qualify as a semilattice. Consider the set {Bob, Susan, Steven, Christina, Thomas}, where Bob and Susan have children Steven, Christina, and Thomas. Letting sup mean the most recent ancestor, inf the most recent descendent, sup{Steven, Chrisitina} doesn’t come as a unique element, and {Steven, Thomas} don’t have an inf. Great try though!

The obstacle above comes as species that have sex. So, if we consider species that don’t have sex, then we’ll have a sup-semilattice. As a concrete example apple trees qualify as a sup-semilattice, in that every two apple trees have a common apple tree ancestor.