Critical trees are neither too short nor too fat

Abstract

We establish lower tail bounds for the height, and upper tail bounds for the width, of critical size-conditioned Bienaymé trees. Our bounds are optimal at this level of generality. We also obtain precise asymptotics for offspring distributions within the domain of attraction of a Cauchy distribution, under a local regularity assumption. Finally, we pose some questions on the possible asymptotic behaviours of the height and width of critical size-conditioned Bienaymé trees.

31 pages, 4 figures, revised version to appear in Annales Henri Lebesgue

References in corpus (4)

• Random trees and applications
• Condensation in nongeneric trees
• A simple proof of Duquesne's theorem on contour processes of conditioned Galton-Watson trees
• A limit theorem for the contour process of conditioned Galton-Watson trees