co.combinatorics - Enumeration and random selection

Suppose there are two finite sets $A$, $B$ and you have some relation on $Atimes B$. If you can randomly sample from $A$ and $B$, you can estimate the average number $a$ of elements of $A$ related to each element of $B$, and the average number $b$ of elements of $B$ related to each element of $A$. Then $a/b=|A|/|B|$, so you can estimate the relative sizes of $A$ and $B$. So if you know the size of $B$ you can estimate the size of $A$. More generally, you can construct a chain of such ratios from the set you want to estimate to one you know the size of.

For example, if you want to estimate the number of trees of order $n$, then consider the trees of order $n-1$ and the relation of removing one leaf from a larger tree to make a smaller tree. Extend this chain to trees of order $n-2$, $n-3$, etc down to order $1$, where you know the number of trees is $1$, and use random sampling to estimate the ratios of the sizes of adjacent classes.