Discrete Mathematics- An Open Introduction - 3rd Edition, 2016a

4.2. Trees 251
one. Let T ′ be the tree resulting from removing v 0 from T (together with
its incident edge). Since we removed a leaf, T ′ is still a tree (the unique
paths between pairs of vertices in T ′ are the same as the unique paths
between them in T).
Now T ′ has k vertices, so by the inductive hypothesis, has k − 1 edges.
What can we say about T? Well, it has one more edge than T ′ , so it has
k edges. But this is exactly what we wanted: v k + 1, e k so indeed
e v − 1. This completes the inductive case, and the proof. qed
There is a very important feature of this induction proof that is worth
noting. Induction makes sense for proofs about graphs because we can
think of graphs as growing into larger graphs. However, this does NOT
work. It would not be correct to start with a tree with k vertices, and then
add a new vertex and edge to get a tree with k + 1 vertices, and note that
the number of edges also grew by one. Why is this bad? Because how do
you know that every tree with k + 1 vertices is the result of adding a vertex
to your arbitrary starting tree? You don’t!
The point is that whenever you give an induction proof that a statement
about graphs that holds for all graphs with v vertices, you must start with
an arbitrary graph with v + 1 vertices, then reduce that graph to a graph
with v vertices, to which you can apply your inductive hypothesis.
Rooted Trees
So far, we have thought of trees only as a particular kind of graph. However,
it is often useful to add additional structure to trees to help solve problems.
Data is often structured like a tree. This book, for example, has a tree
structure: draw a vertex for the book itself. Then draw vertices for each
chapter, connected to the book vertex. Under each chapter, draw a vertex
for each section, connecting it to the chapter it belongs to. The graph will
not have any cycles; it will be a tree. But a tree with clear hierarchy which
is not present if we don’t identify the book vertex as the “top”.
As soon as one vertex of a tree is designated as the root, then every
other vertex on the tree can be characterized by its position relative to the
root. This works because there is a unique path between any two vertices
in a tree. So from any vertex, we can travel back to the root in exactly one
way. This also allows us to describe how distinct vertices in a rooted tree
are related.
If two vertices are adjacent, then we say one of them is the parent of
the other, which is called the child of the parent. Of the two, the parent is
the vertex that is closer to the root. Thus the root of a tree is a parent, but
is not the child of any vertex (and is unique in this respect: all non-root
vertices have exactly one parent).