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Counting binary trees

Binary trees are recursively defined data structures in which each of the non-base instances of the structure contains two substructures of the same sort, ordered with respect to one another (traditionally, as left and right subtrees of the overall structure). We encountered " tipped " binary trees, in which the base instances are " null tips " and each of the non-base instances contains a value as well as the two subtrees, in the reading on mathematical induction. However, there are several variations on this basic theme: Sometimes the base instances are information-bearing values of some sort (numbers, strings, records) rather than null tips, in which case they are likely to be called leaf nodes or simply leaves. Occasionally the non-base binary trees don't contain values directly, just the two subtrees, which in turn may be value-bearing leaves or non-leaf structures. In cases where both the leaves and the non-base tree structures contain values, the values may be of the same type and number, or they may differ. Such structural differences lead to numerous small variations in the algorithms that are used to construct, process, and analyze binary trees, but don't significantly affect the combinatorial characteristics of binary trees that we'll be looking at here. So we'll continue to take tipped binary trees as representative and to use them as our prime example. Suppose that we have a class of values and want to construct a tipped binary tree that contains those values. In how many different ways can such a tree be constructed? Let's start the investigation of this question by looking at the cases in which the cardinality of the class is very small. If the class is empty (so that the cardinality is 0), there is only one tipped binary tree that contains its values, namely the one that is simply a null tip. So, if we use 'T (n)' to denote the number of tipped binary trees containing the members of a class of cardinality n, T (0) = 1.