  The Steiner $k$-eccentricity of a vertex $v$ of a graph $G$ is the maximum
Steiner distance over all $k$-subsets of $V(G)$ which contain $v$. In this
paper Steiner $3$-eccentricity is studied on trees. Some general properties of
the Steiner $3$-eccentricity of trees are given. A tree transformation which
does not increase the average Steiner $3$-eccentricity is given. As its
application, several lower and upper bounds for the average Steiner
$3$-eccentricity of trees are derived.
