The symbolic method of Analytic combinatorics, applied to unlabelled structures. It uses the ordinary generating function.

See here for slides. video.

Elementary identity: $A(z) = \sum_N{N\geq 0} A_N z^N$, where $A_N$ is the number of objects of size $N$

Trees and Catalan numbers

lecture

The number of rooted ordered trees of $n$ nodes is the $n$th Catalan number. Can derive GF by using the fact that "a tree is a node and a sequence of trees". See here.

Can easily extend to binary trees, as done in video

Trees have been related to other combinatorial structures: gambler's ruin sequences, context-free languages, triangulations, ...

Strings

lecture

Powersets and Multisets

Powersets and Multisets