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arXiv · 1703.10652

Most trees are short and fat

Louigi Addario-Berry
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Abstract

This work proves new probability bounds relating to the height, width, and size of Galton-Watson trees. For example, if $T$ is any Galton-Watson tree, and $H$, $W$, and $|T|$ are the height, width, and size of $T$, respectively, then $H/W$ has sub-exponential tails and $H/|T|^{1/2}$ has sub-Gaussian tails. Although our methods apply without any assumptions on the offspring distribution, when information is provided about the distribution the method can be adapted accordingly, and always seems to yield tight bounds.

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Louigi Addario-Berry. 2017-03-30. Most trees are short and fat. https://arxiv.org/abs/1703.10652

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