binary and plane trees are examples of an unlabeled combinatorial structure. trees consist of nodes linked by edges in such a way that there are no cycles. there is generally a node called the root, which has no parent node. in plane trees each node can have an arbitrary number of children. in binary trees, a special case of plane trees, each node can have either two or no children. let formula_17 denote the family of all plane trees. then this family can be recursively defined as follows: