 
Summary: A Family of Random Trees with Random
Edge Lengths*
David Aldous, Jim Pitman
Department of Statistics, University of California, Berkeley, California, 947203860
Received October 16, 1998; accepted 7 May 1999
ABSTRACT: We introduce a family of probability distributions on the space of trees with I
labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive
real lengths. Formulas for distributions of quantities such as degree sequence, shape, and total
length are derived. An interpretation is given in terms of sampling from the inhomogeneous
continuum random tree of Aldous and Pitman (1998). © 1999 John Wiley & Sons, Inc. Random
Struct. Alg., 15, 176195, 1999
Key Words: continuum tree; enumeration; random tree; spanning tree; weighted tree; Cayley's
multinomial expansion
1. INTRODUCTION
A discrete tree is a finite tree in the usual sense of graph theory: n vertices connected
by n  1 undirected edges. A tree with edge lengths is a discrete tree in which each
edge is assigned a strictly positive real number, which we interpret as the length of
the edge. Such trees are often called weighted trees, but we wish to emphasize our
interpretation of the weights as edge lengths. The study of the properties of random
discrete trees, which for uniform models of randomness amounts to enumerations
