Is there tree-based numeral system?

Question

Preferrably such that ((()) ()), (() (())), (((()))), (() () ()) all were different numbers and any tree can be a number.

Answer 1

These trees are known as rooted ordered trees or plane trees and their number linked to Catalan numbers (http://oeis.org/A000108). Paper with such numeral system: A numeral system for the middle levels

Answer 2

One can represent finite trees by a Dyck word, that is a word of the context-free language $D^*$ generated by the grammar $S \to (S) + SS + 1$. The following article gives several possible enumerations for $D^*$:

Zoltan Kasa, Generating and ranking of Dyck words, Acta Univ. Sapientiae, Informatica, 1, 1 (2009) 109–118

See also this paper:

Yu. S. Medvedeva, Fast enumeration of words generated by Dyck grammars, Mathematical Notes 96(1-2):68-83, July 2014