Encoding Scheme For Infinite Set of Symbols: The Percolation Process

It is shown here that the percolation process on binary trees that is equivalent to the critical Galton-Watson process with exactly two offspring for each node, yields a set of infinite symbols equipped with a natural encoding scheme that results in a finite average codeword length as long as the probability of having an offspring p is greater than 1/2 and is smaller than 1/sqrt(2). Furthermore, it is demonstrated that the amount of information encoded is finite.