• 1 Department for Mathematics and Computer Science

We prove a limit theorem for the total Steiner $k$-distance of a random $b$-ary recursive tree with weighted edges. The total Steiner $k$-distance is the sum of all Steiner $k$-distances in a tree and it generalises the Wiener index. The limit theorem is obtained by using a limit theorem in the general setting of the contraction method. In order to use the contraction method we prove a recursion formula and determine the asymptotic expansion of the expectation using the so-called Master Theorem by Roura (2001). In a second step we prove a transformation of the total Steiner $k$-distance of $b$-ary trees with weighted edges to arbitrary recursive trees. This transformation yields the limit theorem for the total Steiner $k$-distance of the linear recursive trees when the parameter of these trees is a non-negative integer.