In this chapter we introduce another interesting infinite group described in terms of functions from ℝ to ℝ, although in this case we only use the closed interval [0, 1]. We begin by discussing dyadic divisions of [0, 1]. A dyadic division of [0, 1] is constructed by first dividing [0, 1] into [0, 1/2] and [1/2, 1], and then proceeding to “pick middles” of the resulting pieces, a finite number of times. For example,

is a dyadic division. We refer to the points of subdivision as the chosen middles, so in the example above the chosen middles are: 1/2, 3/4 and 5/8 (listed in the order they are chosen). We refer to any interval of the form as a standard dyadic interval. Exercise 1 asks you to show that any time you divide [0, 1] into standard dyadic intervals, you necessarily have a dyadic division of [0, 1].

Dyadic divisions of [0, 1] can be encoded using finite, rooted binary trees. Recall from Chapter 6 that a rooted binary tree is a rooted tree where every non-leaf vertex has two descendants.