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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., Comment: 15 pages. Version 2, as of May 22, 2018, includes a proof of Structure Lemma and a few corrections

An m-extracting procedure produces unbiased random bits from a loaded dice with m faces. A binarization takes inputs from an m-faced dice and produce bit sequences to be fed into a (binary) extracting procedure to obtain random bits. Thus, binary extracting procedures give rise to an m-extracting procedure via a binarization. An entropy- preserving binarization is to be called complete, and such a procedure has been proposed by Zhou and Bruck. We show that there exist complete binarizations in abundance as naturally arising from binary trees with m leaves. The well-known leaf entropy theorem and a closely related structure lemma play important roles in the arguments., Comment: 8 pages