Editing Lagrange inversion theorem (section)

===Binary trees===

Consider<ref>{{cite book |last1=Harris|first1= John |last2=Hirst |first2= Jeffry L.| last3= Mossinghoff| first3= Michael |date=2008 |title=Combinatorics and Graph Theory  |publisher= Springer |pages=185–189 |isbn=978-0387797113}}</ref> the set <math>\mathcal{B}</math> of unlabelled [[binary tree]]s. An element of <math>\mathcal{B}</math> is either a leaf of size zero, or a root node with two subtrees.  Denote by <math>B_n</math> the number of binary trees on <math>n</math> nodes.

Removing the root splits a binary tree into two trees of smaller size.  This yields the functional equation on the generating function <math>\textstyle B(z) = \sum_{n=0}^\infty B_n z^n\text{:}</math>
:<math>B(z) = 1 + z B(z)^2.</math>

Letting <math>C(z) = B(z) - 1</math>, one has thus <math>C(z) = z (C(z)+1)^2.</math> Applying the theorem with <math>\phi(w) = (w+1)^2</math> yields
:<math> B_n = [z^n] C(z) = \frac{1}{n} [w^{n-1}] (w+1)^{2n} = \frac{1}{n} \binom{2n}{n-1} =  \frac{1}{n+1} \binom{2n}{n}.</math>

This shows that <math>B_n</math> is the {{mvar|n}}th [[Catalan number]].