The Steiner $k$-eccentricity of a vertex in a graph $G$ is the maximum Steiner distance over all $k$-subsets containing the vertex. The average Steiner $k$-eccentricity of $G$ is the mean value of all vertices’ Steiner $k$-eccentricities in $G$. Let ${\mathbb{T}}_n$ be the set of all $n$-vertex trees, ${\mathbb{T}}_{n,\Delta }$ be the set of $n$-vertex trees with maximum degree $\Delta$, ${\mathbb{T}}_{n,\Delta }^k$ be the set of $n$-vertex trees with exactly $k$ vertices of a given maximum degree $\Delta$, and let ${\mathbb{MT}}_n^k$ be the set of $n$-vertex trees with exactly $k$ vertices of maximum degree. In this paper, we first determine the sharp upper bound on the average Steiner 3-eccentricity of $n$-vertex trees with a given degree sequence. The corresponding extremal graphs are characterized. Consequently, together with majorization theory, all graphs among ${\mathbb{T}}_{n,\Delta }^k$ (resp. ${\mathbb{T}}_{n,\Delta },{\mathbb{MT}}_n^k,{\mathbb{T}}_n$) having the maximum average Steiner 3-eccentricity are identified. Then we characterize the unique $n$-vertex tree with a given segment sequence having the minimum average Steiner 3-eccentricity. Finally, we determine all $n$-vertex trees with a given number of segments having the minimum average Steiner 3-eccentricity.

施泰纳$ķ$- 图中顶点的偏心率$G$是最大的施泰纳距离$ķ$- 包含顶点的子集。平均斯坦纳$ķ$- 偏心率$G$是所有顶点的 Steiner 的平均值$ķ$- 偏心度$G$. 让${\mathbb{吨}}_n$成为所有的集合$n$- 顶点树，${\mathbb{吨}}_{n,\Delta }$是一组$n$- 具有最大度数的顶点树$\Delta$,${\mathbb{吨}}_{n,\Delta }^{ķ}$是一组$n$- 完全具有的顶点树$ķ$给定最大度数的顶点$\Delta$， 然后让${\mathbb{公吨}}_n^{ķ}$是一组$n$- 完全具有的顶点树$ķ$最大度数的顶点。在本文中，我们首先确定了平均 Steiner 3 偏心率的急剧上限$n$-具有给定度数序列的顶点树。表征相应的极值图。因此，连同大数化理论，所有图${\mathbb{吨}}_{n,\Delta }^{ķ}$（分别。${\mathbb{吨}}_{n,\Delta },{\mathbb{公吨}}_n^{ķ},{\mathbb{吨}}_n$) 具有最大平均 Steiner 3 偏心距。然后我们表征独特的$n$-具有给定分段序列的顶点树，具有最小平均 Steiner 3 偏心率。最后，我们确定所有$n$- 具有给定段数的顶点树，具有最小平均 Steiner 3 偏心率。