Chapter 6

Chapter 6. Trees I

6.1 The Tree Abstract Data Type

A tree is an ADT that stores elements hierarchically. With the exception of the top (root) element, each element in a tree has a parent element and zero or more children.

In computer science, a tree is an abstract model of a hierarchical structure
A tree consists of nodes with a parent-child relation
Applications:

Organization charts
File systems
Programming environments

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6.1.1 Terminology and Basic Properties

A tree T is a set of nodes storing elements in a parent-child relationship with the following properties:
- T has a special node r, called the root of T, with no parent node.
- Each node v of T different from r has a unique parent node u.

If node u is the parent of node v, we say that v is a child of u.

Two nodes that are children of the same parent are siblings.

A node is external if has no children, and it is internal if it has one or more children. External nodes are also known as leaves.

The subtree of T rooted at a node v is the tree consisting of all the descendents of v in T (including v itself).

An ancestor of a node is either the parent node itself or an ancestor of the parent of the node.

A node v is a descendent of a node u if u is an ancestor of v.
Example:

Root: node without parent (A)
Internal node: node with at least one child (A, B, C, F)
External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D)
Ancestors of a node (K): parent (F), grandparent (B), grand-grandparent (A), etc.
Depth of a node (F): number of ancestors (2)
Height of a tree: maximum depth (3) of any node (J)
Descendant of a node (A): child (B), grandchild (F), grand-grandchild (I), etc.
Subtree, rooted at a node (C) consists of the node (C) and all its descendents (G, H)

Ordered Trees

A tree is ordered if there is a linear ordering defined for children of each node (siblings ordering).
A binary tree is an ordered tree in which every node has at most two children.
A binary tree is proper if each node has either zero or two children.
For each internal node in a binary tree, we label each child as either being a left child or a right child.
The subtree rooted at a left or right child of an internal node v is called a left subtree or right subtree of v, respectively.

6.1.2 Tree ADT Functions

We use positions to abstract nodes:

Object element(Position)

Generic methods:

integer size()
boolean isEmpty()
ObjectIterator elements()
PositionIterator positions()

Accessor methods:

Position root()
Position parent(Position)
PositionIterator children(Position)

Query methods:

boolean isInternal(Position)
boolean isExternal(Position)
boolean isRoot(Position)

Update methods:

void swapElements(Position, Position)
void replaceElement(Position, Object)

Additional update methods may be defined by data structures implementing the Tree ADT

6.1.3 A Tree Interface
html-6.1a (InspectablePositionalContainer)
html-6.1b (PositionalContainer)
html-6.1c (InspectableTree)
html-6.1d (Tree)

Tree.cpp

6.2 Basic Algorithms on Trees

6.2.1 Running-Time Assumptions

	Input	Output
root()	None	Position	O(1)
parent(v)	Position	Position	O(1)
isInternal(v)	Position	bool	O(1)
isExternal(v)	Position	bool	O(1)
isRoot(v)	Position	bool	O(1)
children(v)	Position	Iterator of positions	O(c_v)
swapElements(v,w)	Two positions	Nine	O(1)
replaceElement(v,e)	A position and an object	None	O(1)
elements()	None	Iterator of objects	O(n)
positions()	None	Iterator of positions	O(n)

6.2.2 Depth and Height

The depth of a node v is the number of ancestors of v, excluding v itself.
html-6.3 (depth)

The height of a node v in a tree T is also defined recursively.

If v is an external node, then the height of v is 0.

Otherwise, the height of v is one plus the maximum height of a child of v.

The height of a tree T is the height of the root of T.
html-6.5 (height1)
html-6.6 (height2)

6.2.3 Preorder Traversal

1. node 2. descendents
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Algorithm preOrder(v)
      visit(v)
     for each child w of v
          preOrder(w)

Tree Printing: An Application of Preorder Traversal
html-6.9 (preorderPrint)

Using Preorder Traversal for Representing a Tree
html-6.10 (parentPrint)

6.2.4 Postorder Traversal

1. descendents 2. node
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Algorithm postOrder(v)
     for each child w of v
          postOrder(w)
     visit(v)

An Illustrative Use of the Postorder Traversal
html-6.12 (postorderPrint)

A Postorder Algorithm for Computing Disk Usage
html-6.13 (diskSpace)

Other Kinds of Traversals

For example, we could traverse a tree so that we visit all the nodes at depth d before we visit the nodes at depth d+1.
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