A family of random trees with random edge lengths

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.

[1]  Montgomery Slatkin,et al.  NULL MODELS FOR THE NUMBER OF EVOLUTIONARY STEPS IN A CHARACTER ON A PHYLOGENETIC TREE , 1991, Evolution; international journal of organic evolution.

[2]  Hosam M. Mahmoud,et al.  Evolution of random search trees , 1991, Wiley-Interscience series in discrete mathematics and optimization.

[3]  Jim Pitman,et al.  The standard additive coalescent , 1998 .

[4]  S. Tavaré,et al.  Line-of-descent and genealogical processes, and their applications in population genetics models. , 1984, Theoretical population biology.

[5]  Jim Pitman.  escent Random Forests , 1999, J. Comb. Theory, Ser. A.

[6]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[7]  Jim Pitman,et al.  Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities , 2000 .

[8]  C. L. Liu.  Generation of trees , 1980, CLAAP.

[9]  J. Moon.  Counting labelled trees , 1970 .

[10]  J. Steele.  Probability theory and combinatorial optimization , 1987 .

[11]  Harry Kesten,et al.  On weighted heights of random trees , 1991 .

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  Jim Pitman,et al.  Partition structures derived from Brownian motion and stable subordinators , 1997 .

[14]  David Aldous,et al.  The Continuum Random Tree III , 1991 .

[15]  H. Pollak,et al.  Steiner Minimal Trees , 1968 .

[16]  L. Alonso,et al.  Random Generation of Trees , 1995, Springer US.

[17]  J. Pitman.  Exchangeable and partially exchangeable random partitions , 1995 .

[18]  Jim Pitman,et al.  Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent , 2000 .