We study the Steiner k-eccentricity on trees, which generalizes the previous one in the paper [X. Li, G. Yu, S. Klavžar, On the average Steiner 3-eccentricity of trees, arXiv:2005.10319, 2020]. To support the algorithm, we achieve much stronger properties for the Steiner k-ecc tree than that in the previous paper. Based on this, a linear time algorithm is devised to calculate the Steiner k-eccentricity of a vertex in a tree. On the other hand, the lower and upper bounds of the average Steiner k-eccentricity index of a tree on order n are established based on a novel technique which is quite different from that in the previous paper but much easier to follow.