The Encoding of Natural Numbers as Nested Parentheses Strings with Associated Probability Distributions

We provide an efficient encoding of the natural numbers {0,1,2,3,...} as strings of nested parentheses {(),(()),(()()),((())),...}, or considered inversely, an efficient enumeration of such strings. The technique is based on the recursive definition of the Catalan numbers. The probability distributions arising from this encoding are explored. Applications of this encoding to prefix-free data encoding and recursive function theory are briefly considered.