Combinatorics Through Guided Discovery, 2004a

2.3. Graphs and Trees 45
(c) Can you tell from the sequence of b i s what a 1 is? (h)
(d) Find a bijection between labelled trees and something you can “count”
that will tell you how many labelled trees there are on n labelled
vertices. (h)
The sequence b 1 , b 2 ,...,b n−2 in Problem 111 is called a Prüfer coding or Prüfer
code for the tree. There is a good bit of interesting information encoded into the
Prüfer code for a tree.
Problem 113. What can you say about the vertices of degree one from the
Prüfer code for a tree labeled with the integers from 1 to b? (h)
Problem 114. What can you say about the Prüfer code for a tree with exactly
two vertices of degree 1? (and perhaps some vertices with other degrees as
well)? Does this characterize such trees?
⇒
Problem 115. What can you determine about the degree of the vertex labelled
i from the Prüfer code of the tree? (h)
⇒
Problem 116. What is the number of (labelled) trees on n vertices with three
vertices of degree 1? (Assume they are labelled with the integers 1 through
n.) This problem will appear again in the next chapter after some material
that will make it easier. (h)
2.3.4 Spanning trees
Many of the applications of trees arise from trying to find an efficient way to connect
all the vertices of a graph. For example, in a telephone network, at any given time
we have a certain number of wires (or microwave channels, or cellular channels)
available for use. These wires or channels go from a specific place to a specific
place. Thus the wires or channels may be thought of as edges of a graph and the
places the wires connect may be thought of as vertices of that graph. A tree whose
edges are some of the edges of a graph G and whose vertices are all of the vertices
of the graph G is called a spanning tree of G. A spanning tree for a telephone
network will give us a way to route calls between any two vertices in the network.
In Figure 2.3.3 we show a graph and all its spanning trees.