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Invariant Galton–Watson trees: metric properties and attraction with respect to generalized dynamical pruning

Published online by Cambridge University Press:  12 January 2023

Yevgeniy Kovchegov*
Affiliation:
Oregon State University
Guochen Xu*
Affiliation:
Oregon State University
Ilya Zaliapin*
Affiliation:
University of Nevada Reno
*
*Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA.
*Postal address: Department of Mathematics, Oregon State University, Corvallis, OR 97331-4605, USA.
****Postal address: Department of Mathematics and Statistics, University of Nevada Reno, Reno, NV 89557-0172, USA. Email address: zal@unr.edu

Abstract

The invariant Galton–Watson (IGW) tree measures are a one-parameter family of critical Galton–Watson measures invariant with respect to a large class of tree reduction operations. Such operations include the generalized dynamical pruning (also known as hereditary reduction in a real tree setting) that eliminates descendant subtrees according to the value of an arbitrary subtree function that is monotone nondecreasing with respect to an isometry-induced partial tree order. We show that, under a mild regularity condition, the IGW measures are attractors of arbitrary critical Galton–Watson measures with respect to the generalized dynamical pruning. We also derive the distributions of height, length, and size of the IGW trees.

Type
Original Article
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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