Enumerating k-Way Trees

This paper makes a contribution to the enumeration of trees. We prove a new result about k-way trees, point out some special cases, use it to give a new proof of Cayley’s enumeration formula for labelled trees, and observe that our techniques allow various types of k-way trees to be generated uniformly at random. We recall the definition: a k-way tree is either empty or it consists of a root node and a sequence of k k-way subtrees. These structures are often used in software systems to store data, with k = 2 (binary trees) being the commonest case. It is usual to represent a k-way tree by a diagram in which the root node is connected to its non-empty subtrees by edges which point in one of k fixed directions. Fig. 1 depicts a 3-way tree with 5 directions of the first type, 4 of the second type and 3 of the third type.