A Note on Approximation of Uniform Distributions from Variable-to-Fixed Length Codes

In this correspondence, we prove that the probability distri- bution induced on the leaves of a Tunstall parse tree for a given source is a (unique) lower bound in the partially ordered set of the probability distributions induced by all possible parse trees with a same number of leaves, and ordered according to the majorization partial order. We apply this result to the problem of optimally approximating a uniform distribution with flips of a biased coin.