Def/Associative

 =Regrouping= Suppose that $X$ is a set, and $\cdot$ is an associative binary operation on $X$. The most important application of associativity is the ability to defines::regroup large expressions involving $\cdot$. More precisely, we have the following result:


 * Suppose that $T$ and $T'$ are rooted ordered binary trees, with leaves labeled by $X$, and the same number of leaves. Assume that the resulting finite sequences of leaves, $(t_1, \ldots, t_n)$ and $(t_1', \ldots, t_n')$ are equal.  Then $\prod T = \prod T'$.

As a result, one often writes $\prod_{i=1}^n t_i$ rather than $\prod T$ (or $\prod T'$), since product is independent of the tree whose sequence of leaves is $(t_1, \ldots, t_n)$.

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