On the Enumeration of Tree-Rooted Maps

Abstract

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).