DISCRETE MATHEMATICS AND ITS APPLICATIONS
by Kenneth H. Rosen
Chapter 9 -- Trees
9.1 Introduction
A tree is a connected undirected graph with no simple circuits (cycles).
An undirected graph is a tree iff there is a unique simple path between any two of its
vertices.
A rooted tree is a directed graph constructed from a tree with one vertex designated as a
root, and each edge directed away from the root.
In a rooted tree T, let (u, v) be a directed edge in T, then u is the parent of v, and v the child
of u; vertices with the same parents are called siblings; the ancestors of a vertex v other than
the root are the vertices in the path from the root to v; the descendants of a vertex v are those
vertices that have v as an ancestor; a leaf is a vertex with no children; vertices that have
children are called internal vertices; a vertex v together with its descendants and all edges
incident to these descendants form a subtree.
A rooted tree where every internal vertex has m children is called an m-ary tree; it is a
binary tree if m = 2.
The level of a vertex v in a rooted tree is the length of the unique path from the root to v.
The height of a rooted tree is the maximum of the levels of vertices.
A rooted m-ary tree of height h is called balanced if all leaves are at levels h or h-1.
In an ordered rooted tree the children of each internal vertex are ordered. If a vertex has
two children, the first child is called the left child and the second child is called the right
child. The tree rooted at the left child is called the left subtree, and that rooted at the right
child is called the right subtree.
Trees may be used to model:
Saturated hydrocarbons
Organizations
File directories
Interconnection networks for parallel processing
Properties of trees
A tree with n vertices has exactly n 1 edges.
A full m-ary tree with i internal vertices contains n = mi + 1 vertices.
A full m-ary tree with
n vertices has i = (n1)/m internal vertices and l = [(m1)n + 1]/m leaves.
i internal vertices has n = m i + 1 vertices and l = (m1)i + 1 leaves.
l leaves has n = (ml1)/(m1) vertices and i = (l1)/(m1) internal vertices.
There are at most mh leaves in an m-ary tree of height h.
If an m-ary tree of height h has l leaves, then h logm l . If the m-ary tree is full and
balanced, then h = logm l .
Exercises in 9.1
