Summary: 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 can be applied to study the random minimum-weight spanning tree question, when the edge-weight distribution is allowed to vary almost arbitrarily with \(n\).