The set of objects studied in discrete mathematics can be finite or infinite. In real-world applications, the set of objects of interest are mainly finite, the study of which is often called finite mathematics. In some mathematics curricula, the term "finite mathematics" refers to courses that cover discrete mathematical concepts for business, while "discrete mathematics" courses emphasize discrete mathematical concepts for computer science majors.
A 1-forest (a maximal pseudoforest), formed by three 1-trees
In graph theory, a pseudoforest is an undirected graph in which every connected component has at most one cycle. That is, it is a system of vertices and edges connecting pairs of vertices, such that no two closed paths of consecutive edges share any vertex with each other, nor can any two such closed paths be connected to each other by a path of consecutive edges. A pseudotree is a connected pseudoforest.
The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan attribute the naming of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – they have very few edges relative to their number of vertices – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests.