  We study a new type of random minimum spanning trees. It is built on the
complete graph where each vertex is given a weight, which is a positive real
number. Then, each edge is given a capacity which is a random variable that
only depends on the product of the weights of its endpoints. We then study the
minimum spanning tree corresponding to the edge capacities. Under a condition
of finite moments on the node weights, we show that the expected diameter and
typical distances of this minimum spanning tree are of order $n^{1/3}$. This is
a generalization of the results of Addario-Berry, Broutin, and Reed [2009]. We
then use our result to answer a conjecture in statistical physics about typical
distances on a closely related object. This work also sets the ground for
proving the existence of a non-trivial scaling limit of this spanning tree (a
generalization of the result of Addario-Berry, Broutin, Goldschmidt, and
Miermont [2017])). Our proof is based on a detailed study of rank-1 critical
inhomogeneous random graphs, done in Safsafi [2020], and novel couplings
between exploration trees related to those graphs and Galton-Watson trees.
