On the number of Sperner vertices in a tree

A vertex $v$ of a tree $T$ is called a Sperner vertex if the in-tree $T(v)$ obtained from $T$ by orientation of all edges towards $v$ has the Sperner property, i.e. there exists a largest subset $A$ of mutually unreachable vertices in it such that all vertices in $A$ are equidistant to $v$. Some explicit methods to count the number of Sperner vertices in certain special trees are presented.