Reversibility and Universality by Andrew Adamatzky

Author:Andrew Adamatzky
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The function can be calculated recursively:

(5)

The term of the clause is zero, which implies that there is no binary tree that is smaller than the empty tree. The clause returns the sum of the numbers of the binary trees that satisfy(a) ,

(b) and , and

(c) and ,

respectively. The size of the left subtrees that satisfy condition (a) lies in the range of . The first term is the sum of the sizes, i.e., for the range. The second term is the product of the number of left subtrees that satisfy and the number of possible right subtrees corresponding to the left subtrees. The third term is the number of right subtrees that satisfy . The conditions (a), (b), and (c) are mutually exclusive, and the sum of the binary trees that satisfy the conditions is equal to the number of binary trees of the same size as T, which are smaller than T in the natural order.

For example, the binary tree in Fig. 2 has a left subtree of rank zero and a right subtree of rank seven from Fig. 1. Therefore, the rank of the binary tree in Fig. 2 is .

Fig. 2Binary tree of rank 49 (corresponding to the tree permutation 104235)

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