Combinatorics Through Guided Discovery, 2004a

44 2. Applications of Induction and Recursion in Combinatorics and Graph Theory
· Problem 109. What is the minimum number of vertices of degree one in a
finite tree? What is it if the number of vertices is bigger than one? Prove
that you are correct. (h)
⇒ ·
Problem 110. In a tree, given two vertices, how many paths can you find
between them? Prove that you are correct.
⇒∗
Problem 111. How many trees are there on the vertex set {1, 2}? On the
vertex set {1, 2, 3}? When we label the vertices of our tree, we consider the
tree which has edges between vertices 1 and 2 and between vertices 2 and 3
different from the tree that has edges between vertices 1 and 3 and between
2 and 3. See Figure 2.3.2.
1 2
3
2 3
1
2 1
3
Figure 2.3.2: The three labelled trees on three vertices
How many (labelled) trees are there on four vertices? You don’t have a lot
of data to guess from, but try to guess a formula for the number of labelled
trees with vertex set {1, 2, ··· , n}. (h)
We are now going to introduce a method to prove the formula you guessed.
Given a tree with two or more vertices, labelled with positive integers, we define a
sequence b 1 , b 2 ,...of integers inductively as follows: If the tree has two vertices,
the sequence consists of one entry, namely the label of the vertex with the larger
label. Otherwise, let a 1 be the lowest numbered vertex of degree 1 in the tree. Let
b 1 be the label of the unique vertex in the tree adjacent to a 1 and write down b 1 .
For example, in the first graph in Figure 2.3.1, a 1 is 1 and b 1 is 2. Given a 1 through
a i−1 , let a i be the lowest numbered vertex of degree 1 in the tree you get by deleting
a 1 through a i−1 and let b i be the unique vertex in this new tree adjacent to a i .For
example, in the first graph in Figure 2.3.1, a 2 =2and b 2 =3. Then a 3 =5and
b 3 =4. We use b to stand for the sequence of b i s we get in this way. In the tree (the
first graph) in Figure 2.3.1, the sequence b is 2344378. (If you are unfamiliar with
inductive (recursive) definition, you might want to write down some other labelled
trees on eight vertices and construct the sequence of b i s.)
Problem 112.
(a) How long will the sequence of b i s be if it is computed from a tree with
n vertices (labelled with 1 through n)?
(b) What can you say about the last member of the sequence of b i s? (h)