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The binary operator for vectors is defined such that <math>\begin{pmatrix} e_i^0 \\ \vdots \\ e_i^{m-1}\end{pmatrix} \oplus^\star \begin{pmatrix} e_j^0 \\ \vdots \\ e_j^{m-1}\end{pmatrix} = \begin{pmatrix} e_i^0 \oplus e_j^0 \\ \vdots \\ e_i^{m-1} \oplus e_j^{m-1} \end{pmatrix}</math>. The algorithm further assumes that in the beginning <math>x_i = v_i</math> for all <math>i</math> and <math>p</math> is a power of two and uses the processing units <math>p_0,p_1,\dots p_{n-1}</math>. In every iteration, half of the processing units become inactive and do not contribute to further computations. The figure shows a visualization of the algorithm using addition as the operator. Vertical lines represent the processing units where the computation of the elements on that line take place. The eight input elements are located on the bottom and every animation step corresponds to one parallel step in the execution of the algorithm. An active processor <math>p_i</math> evaluates the given operator on the element <math>x_i</math> it is currently holding and <math>x_j</math> where <math>j</math> is the minimal index fulfilling <math>j > i</math>, so that <math>p_j</math> is becoming an inactive processor in the current step. <math>x_i</math> and <math>x_j</math> are not necessarily elements of the input set <math>X</math> as the fields are overwritten and reused for previously evaluated expressions. To coordinate the roles of the processing units in each step without causing additional communication between them, the fact that the processing units are indexed with numbers from <math>0</math> to <math>n-1</math> is used. Each processor looks at its <math>k</math>-th least significant bit and decides whether to get inactive or compute the operator on its own element and the element with the index where the <math>k</math>-th bit is not set. The underlying communication pattern of the algorithm is a binomial tree, hence the name of the algorithm.
Only <math>p_0</math> holds the result in the end, therefore it is the root processor. For an
==== Runtime analysis ====
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