  Peres algorithm applies the famous von Neumann trick recursively to produce
unbiased random bits from biased coin tosses. Its recursive nature makes the
algorithm simple and elegant, and yet its output rate approaches the
information-theoretic upper bound. However, it is relatively hard to explain
why it works, and it appears partly due to this difficulty that its
generalization to many-valued source was discovered only recently. Binarization
tree provides a new conceptual tool to understand the innerworkings of the
original Peres algorithm and the recently-found generalizations in both aspects
of the uniform random number generation and asymptotic optimality. Furthermore,
it facilitates finding many new Peres-style recursive algorithms that have been
arguably very hard to come by without this new tool.
