# Trees

```CSCI 3333 Data Structures
Trees
by
Dr. Bun Yue
Professor of Computer Science
yue@uhcl.edu
http://sce.uhcl.edu/yue/
2013
Acknowledgement
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Mr. Charles Moen
Dr. Wei Ding
Ms. Krishani Abeysekera
Dr. Michael Goodrich
What is a Tree
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
In computer
Computers”R”Us
science, a tree is
an abstract
model of a
Sales
Manufacturing
R&amp;D
hierarchical
structure
A tree consists US International Laptops Desktops
of nodes with a
parent-child
Europe
Asia
relation
Trees

Each node in the tree has

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0 or 1 parent node.
0 or more child nodes.
Only the root node has no parent.
A tree can be considered as a special case
of a directed (acyclic) graph.
Example: a family tree may not
necessarily be a tree in the computing
sense.
Trees
British Royal Family Tree
Queen Victoria
King Edward VII
Albert Victor
Alice
King George V
King Edward VIII
King George VI
Queen Elizabeth II
Charles
William
Henry
Ann
Peter
Andrew
Zara
Beatrice
Eugenie
Edward
5
Some Tree Applications

Applications:

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Organization charts
File Folders
Email Repository
Programming environments: e.g.
Recursive Definition of Trees
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An empty tree is a tree.
A tree contains a root node
connected to 0 or more child subtrees.
Trees
Are these trees?
1
3
2
4
8
Tree Terminology
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Root: node without parent (A)
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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: parent,
grandparent, grandgrandparent, etc.
Depth of a node: number of
E
ancestors
Height of a tree: maximum
depth of any node (3)
Descendant of a node: child,
I
grandchild, grand-grandchild,
etc.
Subtree: tree consisting
of a node and its
descendants
A
B
C
F
J
G
K
D
H
subtree

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We use positions to
abstract nodes
Generic methods:
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integer size()
boolean isEmpty()
Iterator elements()
Iterator positions()
Accessor methods:
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position root()
position parent(p)
positionIterator
children(p)
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Query methods:
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Update method:
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boolean isInternal(p)
boolean isExternal(p)
boolean isRoot(p)
object replace (p, o)
methods may be defined
by data structures
implementing the Tree
Interface TreeNode in Java
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Enumeration children(): Returns the children of
boolean getAllowsChildren(): Returns true if the
TreeNode getChildAt(int childIndex): Returns the
child TreeNode at index childIndex.
int getChildCount(): Returns the number of
int getIndex(TreeNode node): Returns the index
of node in the receivers children.
TreeNode getParent(): Returns the parent
Boolean isLeaf(): Returns true if the receiver is a
leaf.
Interface MutatableTreeNode
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Sub-interface of TreeNode
Methods:
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void insert(MutableTreeNode child, int index):
void remove(int index): Removes the child at
void remove(MutableTreeNode node):
void removeFromParent(): Removes the
void setParent(MutableTreeNode newParent):
Sets the parent of the receiver to newParent.
void setUserObject(Object object): Resets the
user object of the receiver to object.
Size of a tree
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Size = number of nodes in the tree.
Recursion analysis:
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Base case: empty root =&gt; return 0;
Recursive case: non-empty root
=&gt; return 1 + sum of size of all
child sub-trees.
Size of a tree
Algorithm size(TreeNode root)
Input TreeNode root
Output: size of the tree
if (root = null)
return 0;
else
return 1 + Sum of sizes of all children of
root
end if
Implementation in Java
public static int size(TreeNode root) {
if (root == null) return 0;
Enumeration enum = root.children();
int result = 1; // count the root.
while (enum.hasMoreElement()) {
result += size((TreeNode)
enum.nextElement());
}
return result;
}
Trees in languages
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There are no general tree classes in the
standard libraries in many languages:
e.g. C++ STL, Ruby, Python, etc.
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No single interface tree design satisfies all
users.
There may be specialized trees, especially
for GUI design. E.g.
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JTree in Java
FXTreeList in Ruby
Tree Traversal
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Traversing is the systematic way of
accessing, or ‘visiting’ all the nodes
in a tree.
Consider the three operations on a
binary tree:
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V: Visit a node
L: (Recursively) traverse the left
subtree of a node
R: (Recursively) traverse the right
subtree of a node
Tree Traversing
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We can traverse a tree in six
different orders:
LVR
VLR
LRV
VRL
RVL
RLV
Preorder Traversal
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A traversal visits the nodes of
a tree in a systematic manner
In a preorder traversal, a
node is visited before its
descendants
Application: print a structured
document
1
Algorithm preOrder(v)
visit(v)
for each child w of v
preorder (w)
Make Money Fast!
2
5
1. Motivations
9
2. Methods
3
4
1.1 Greed
1.2 Avidity
6
2.1 Stock
Fraud
7
2.2 Ponzi
Scheme
References
8
2.3 Bank
Robbery
Java Implementation
public static void preorder(TreeNode root) {
if (root != null) {
visit(root); // assume declared.
Enumernation enum = root.children();
while (enum.hasMoreElement()) {
preorder((TreeNode)
enum.nextElement());
}
}
Postorder Traversal
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In a postorder traversal, a
node is visited after its
descendants
Application: compute space
used by files in a directory
and its subdirectories
9
Algorithm postOrder(v)
for each child w of v
postOrder (w)
visit(v)
cs16/
3
8
7
homeworks/
todo.txt
1K
programs/
1
2
h1c.doc
3K
h1nc.doc
2K
4
DDR.java
10K
5
Stocks.java
25K
6
Robot.java
20K
Inorder Traversal
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For binary tree, inorder traversal
can also be defined: L V R.
Java Implementation
public static void inorder(TreeNode
root) {
if (root != null) {
inorder(root.getChildAt(0));
visit(root); // assume declared.
inorder(root.getChildAt(1));
}
Expression Notation
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There are three common notations
for expressions:
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Prefix: operator come before operands.
Postfix: operator come after operands.
Infix:
Binary operator come between operands.
 Unary operator come before the operand.
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Examples
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Infix: (a+b)*(c-d/f/(g+h))
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Prefix: *+ab-c//df+gh
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Used by human being.
E.g. functional language such as Lisp:
Lisp: (* (+ a b) (- c (/ d (/ f (+ g h)))))
Postfix: ab+cdf/gh+/-*
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E.g. Postfix calculator
Expression Trees
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An expression tree can be used to
represent an expression.
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Operator: root
Operands: child sub-trees
Variable and constants: leaf nodes.
Traversals of Expression Trees
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Preorder traversal =&gt; prefix
notation.
Postorder traversal =&gt; postfix
notation.
Inorder traversal with parenthesis
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A node is represented
by an object storing
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Element
Parent node
Sequence of children
nodes

B
Node objects implement
D
C

A
B
A

D
F
F

E
C

E
Tree Classes in Ruby
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FXTreeItem: a tree node for GUI
application.
Some methods:
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Tree related: childOf?, parentOf?,
numChildren, parent, etc.
Traversal of child nodes: first, next,
last, etc.
GUI related: selected?, selected=,
hasFocus?, etc.
Tree Classes in Ruby 2
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FXTreeList: a list of FXTreeItem.
Include methods for list
manipulation, such as appendItem,
clearItems, etc.