# Most trees are short and fat

## TL;DR

This paper establishes new probability bounds for the height, width, and size of Galton-Watson trees, showing that certain ratios have sub-exponential or sub-Gaussian tail behaviors, applicable broadly without distribution assumptions.

## Contribution

It introduces general probability bounds for Galton-Watson trees' dimensions, adaptable to specific offspring distributions, improving understanding of their typical shapes.

## Key findings

- H/W ratio has sub-exponential tails
- H/|T|^{1/2} ratio has sub-Gaussian tails
- Bounds are tight and distribution-agnostic

## 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.