 
Summary: A Random Tree Model Associated
with Random Graphs
David Aldous*
Department of Statistics, University of California, Berkeley CA 94720
ABSTRACT
Grow a tree on n vertices by starting with no edges and successivelyadding 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 minimumweight spanning tree question, when the edgeweight distribution is
allowed to vary almost arbitrarily with n.
1. INTRODUCTION
The construction indicated in the abstract, and stated more formally in Section 2,
yields a certain random tree F,,on n vertices. It is easy to calculate that, for the
star graph t, centered at vertex 1,
P(F" = I,) = 2"'(n  1)!/(2n  2)!
and so for n 1.4, F,, is not the uniform random labeled tree (for which the
probability is 1ln"2). Other explicit calculations are harder. The substance of
this paper is Theorem 1 below, which gives information about certain aspects of
