Limits of functions on trees

We present various formulations for the limit of a function from a tree to the reals.\ud The formulations will not only generalize the notion of limit of a sequence of reals,\ud but will also have potential applications in obtaining upper and lower bounds for\ud the Hausdorff dimension of certain classes of Cantor sets. One formulation will\ud involve the branches of the tree, another the maximal antichains, and the third\ud will involve the nonstandard concept of a slice. Our branch formulation for limit\ud will be the simplest to define, while the other two formulations will require us to\ud make both the set of maximal antichains and the set of slices into directed sets. We\ud then study and compare these and notice that they do not always agree (in value),\ud although we do get some nice inequalities. While investigating the interactions\ud amongst the branches, maximal antichains, and slices, we shall see that we can get\ud equality between some or all of our formulations by placing different sorts of mild\ud restrictions on the function or the tree. We end by discussing a particular class of\ud trees, called the omega-trees, which are useful for representating Cantor sets, and\ud then use our observations to reveal a more constructive proof for one of our earlier\ud lemmas restricted to the class of pruned omega-trees