instance of self-organized criticality.

The invasion cluster has some properties in common with the IIC but not all. For example, results of Kesten [@kesten-IIC] and Zhang [@zhang-invasion] show that the growth exponents of the two are equal on the two-dimensional lattice; however the measures of the two clusters are mutually singular on the lattice [@damron-invasion] as well as on a regular tree [@angel-invasion2008]. Our focus is the comparison of the laws induced on paths by both the IIC and invasion percolation.

On a Galton-Watson tree $T$, there is a natural measure on paths, the limit-uniform measure $\mu_T$, which although it does not restrict precisely to the uniform measure on each generation, approximates this as closely as possible. There is not, however, a fast algorithm for sampling from it. Rules such as “split equally at each node” lead to rapid sampling but the wrong entropy; in other words, the Radon-Nikodym derivative with respect $\mu_T$ on generation $N$ will be exponential in $N$. It is not hard to show that on almost every Galton-Watson tree (assuming a $Z \log Z$ moment for the offspring distribution), the unique path in the IIC has law $\mu$. Since sampling from the IIC is problematic, it 