BIJECTIONS BETWEEN CERTAIN FAMILIES OF LABELLED AND UNLABELLED d-ARY TREES

We are dealing here with d-ary trees, which are amongst the most fundamental tree structures with applications, e.g., in combinatorics, computer science and biology, see, e.g., [1,3–5,9]. A d-ary tree is either an empty tree or it consists of a root node, to which an ordered sequence of exactly d subtrees is attached that are itself d-ary trees. We denote the family of d-ary trees by Td and the empty tree by ε. In particular in the computer science related literature one sometimes uses instead of the symbol ε the notion of external nodes with a certain symbol distinguishable from the proper nodes, called internal nodes. Each node v in the tree has then exactly d children attached to v and we may speak of the first child, . . . , the d-th child, where we have to allow “empty children” ε. This recursive description can also be expressed via the following formal equation for Td, where ∪̇ denotes the disjoint union of two combinatorial families: