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On the number of leaves of a euclidean minimal spanning tree

Published online by Cambridge University Press:  14 July 2016

J. Michael Steele ,
Lawrence A. Shepp  and
William F. Eddy
J. Michael Steele*
Affiliation:
Princeton University
Lawrence A. Shepp*
Affiliation:
AT&T Bell Laboratories
William F. Eddy*
Affiliation:
Carnegie-Mellon University
*
∗Postal address: Program in Engineering Statistics, Princeton University, Princeton, NJ 08544, USA.
∗∗Postal address: AT&T Bell Laboratories, Murray Hill, NJ 07974, USA.
∗∗∗Postal address: Department of Statistics, Carnegie-Mellon University, Pittsburgh, PA 15213, USA.
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Abstract

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

Keywords

SUBADDITIVE EUCLIDEAN FUNCTIONALSVERTEX DEGREESSELF-SIMILAR PROCESSESEFRON–STEIN INEQUALITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 4 , December 1987 , pp. 809 - 826
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

Research supported in part by National Science Foundation Grant DMS-8414069.

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