Section 5.2: Trees and their representations

Abstract:

Trees are defined, some applications are presented, some computer representation strategies are considered, and tree traversal algorithms are discussed.

Tree Definition and Terminology

Definition: a tree is an acyclic, connected graph with one node designated as the root node.

"Because a tree is a connected graph, there is a path from the root to any other node in the tree; because the tree is acyclic, that path is unique [provided that no arc is used twice]." p. 364, Gersting.

The set of trees can also be defined recursively, as follows:

• Base case: A single node is a tree, with that node as root.
• Inductive step: The graph formed by attaching a new node r by a single arc to each root of trees is a tree.

Tree terminology

Exercise #2, p. 374.

Examples of trees in action (p. 365)

• My favorite: the UNIX operating system (with root node called root!)
• The Hexstat probability demonstrator (a full binary tree)
• Family ``trees'' (if intermarriages are forbidden)
• Org charts (most people report to only one person above them - if not, a cycle results, and it's not a tree!)

Tree Representations

Binary trees (e.g. the Hexstat probability demonstrator) are special in that their children are classified as left or right. So in order to represent them, we need to specify left and right children for each node. Two representations are shown in Example 23, p. 367: a two-column array, and an adjacency list with three-field records.

Practice 20, p. 367.

Tree Traversal Algorithms

We'll look at three, which are represented recursively as follows:

Exercise #18, p. 377.

Each traversal method has advantages in different situations. The example we consider is binary trees as representations of arithmetic expressions.

• Inorder traversal writes the expressions in the most familiar form from school (Infix notation). Inorder traversal requires the introduction of parentheses to make the meaning of the expression unique.
• Postorder traversal allows us to eliminate nodes once traversed (if we want to deallocate storage, for example). This is called Reverse Polish Notation, named for the Pole Jan Lukasiewicz.
• Preorder traversal represents the expressions in Polish notation (which is what LISP uses (God's computer language)). Like postorder, preorder doesn't require parentheses.

Practice 22, p. 372.

Exercise #5, p. 374.