Encoding of binary trees as a regular language?
As Peter Shor mentions in a comment, the number of full binary trees with $n+1$ leaves is given by the $n$-th Catalan number $C_n$. It is known that the Catalan numbers have generating function

$C(x) = \frac{1-\sqrt{1-4x}}{2x}$.

On the other hand, it is known (Chomsky, Schützenberger; 1963) that regular languages have rational generating functions.

Clearly, $C$ is not rational. This implies that there is no regular encoding of full binary trees where each tree has a unique representation.

(Note that the generating function of arbitrary binary trees is not rational, either.)