A hierarchy on a set $S$ is a collection $H$ of subsets of $S$ for which for every pair $A,B$ of elements of $H$, either $A$ contains $B$ or $B$ contains $A$ or the intersection of $A$ and $B$ is empty. Additionally, a hierarchy on $S$ contains $S$ and all singleton subsets of $S$. Hierarchies are also known as laminar families and total partitions. A random hierarchy $H$ on a countable set $S$...