A random tree model associated with random graphs

Abstract

Abstract Grow a tree on n vertices by starting with no edges and successively adding an edge chosen uniformly from the set of possible edges whose addition would not create a cycle. This process is closely related to the classical random graph process. We describe the asymptotic structure of the tree, as seen locally from a given vertex. In particular, we give an explicit expression for the asymptotic degree distribution. Our results an be applied to study the random minimum‐weight spanning tree question, when the edge‐weight distribution is allowed to vary almost arbitrarily with n .