Spanning Trees

In this chapter, we will study trees in considerably more detail than in the introductory Section 1.2. Beginning with some further characterizations of trees, we then present another way of determining the number of trees on n vertices which actually applies more generally: it allows us to compute the number of spanning trees in any given connected graph. The major part of this chapter is devoted to a network optimization problem, namely to finding a spanning tree for which the sum of all edge lengths is minimal. This problem has many applications; for example, the vertices might represent cities we want to connect to a system supplying electricity; then the edges represent the possible connections and the length of an edge states how much it would cost to build that connection. Other possible interpretations are tasks like establishing traffic connections (for cars, trains or planes: the connector problem) or designing a network for TV broadcasts. Finally, we consider Steiner trees (these are trees where it is allowed to add some new vertices) and arborescences (directed trees).