Trees and forests

Abstract

The basic terminology of graphs, as vertex-edge pairs is presented, leading to the definitions of trees, in the sense of rooted trees, and unrooted, or free, trees. It is shown how to build up trees from the tree with a single vertex, using the beta-product and the B+ operation, together with a prefix (Polish) operator. Formal linear combinations of forests are introduced as the “forest space” and, as an application, the enumeration of trees is studied. Partitions of sets and numbers are introduced and related to trees. The evolution of partitions and of trees, is discussed as the complexity is increased step by step. Subtrees and prunings are introduced and analysed. The antipode is introduced and its properties discussed, including the involution property. The interplay between groups, linear algebra and trees is briefly investigated. Truncated trees and forests are introduced. This leads to the algebra of linear combinations of univalent stumps which act as linear operators on the tree space.