  The Steiner $k$-eccentricity of a vertex in graph $G$ is the maximum Steiner
distance over all $k$-subsets containing the vertex. %Some general properties
of the Steiner 3-eccentricity of trees are given. Let $\mathbb{T}_n$ be the set
of all $n$-vertex trees, $\mathbb{T}_{n,\Delta}$ be the set of $n$-vertex trees
with given maximum degree equal to $\Delta$, $\mathbb{T}_n^k$ be the set of
$n$-vertex trees with exactly $k$ vertices of maximum degree, and let
$\mathbb{T}_{n,\Delta}^k$ be the set of $n$-vertex trees with exactly $k$
vertices of given maximum degree equal to $\Delta.$ In this paper, we first
determine the sharp upper bound on the average Steiner 3-eccentricity of
$n$-vertex trees with given degree sequence. The corresponding extremal graph
is characterized. Consequently, together with majorization theory, the unique
graph among $\mathbb{T}_n$ (resp. $\mathbb{T}_{n,\Delta}$, $\mathbb{T}_n^k,
\mathbb{T}_{n,\Delta}^k$) having the maximum average Steiner 3-eccentricity is
identified. Then we characterize the unique $n$-vertex tree with given segment
sequence having the largest average Steiner 3-eccentricity. Similarly, the
$n$-vertex tree with given number of segments having the largest average
Steiner 3-eccentricity is determined.
