Trees

by

Dr. Bun YueProfessor of Computer Science

yue@uhcl.eduhttp://sce.uhcl.edu/yue/

2013

CSCI 3333 Data Structures

Acknowledgement

Mr. Charles Moen Dr. Wei Ding Ms. Krishani Abeysekera Dr. Michael Goodrich

What is a Tree In computer

science, a tree is an abstract model of a hierarchical structure

A tree consists of nodes with a parent-child relation

Computers”R”Us

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Laptops DesktopsUS International

Europe Asia Canada

Trees

Each node in the tree has 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.

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British Royal Family TreeTrees

Queen Victoria

King Edward VII

Albert Victor

Alice

King George V

King George VIKing Edward VIII

Queen Elizabeth II

Charles Ann Andrew Edward

William Henry Peter Zara Beatrice Eugenie

Some Tree Applications

Applications: Organization charts File Folders Email Repository Programming environments: e.g. drop

down menus.

Recursive Definition of Trees

An empty tree is a tree. A tree contains a root node

connected to 0 or more child sub-trees.

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Are these trees?Trees

1 2

34

subtree

Tree Terminology 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: parent, grandparent, grand-grandparent, etc.

Depth of a node: number of ancestors

Height of a tree: maximum depth of any node (3)

Descendant of a node: child, grandchild, grand-grandchild, etc.

A

B DC

G HE F

I J K

Subtree: tree consisting of a node and its descendants

Tree ADT (§ 6.1.2) We use positions to

abstract nodes Generic methods:

integer size() boolean isEmpty() Iterator elements() Iterator positions()

Accessor methods: position root() position parent(p) positionIterator

children(p)

Query methods: boolean isInternal(p) boolean isExternal(p) boolean isRoot(p)

Update method: object replace (p, o)

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

Interface TreeNode in Java

Enumeration children(): Returns the children of the receiver as an Enumeration.

boolean getAllowsChildren(): Returns true if the receiver allows children.

TreeNode getChildAt(int childIndex): Returns the child TreeNode at index childIndex.

int getChildCount(): Returns the number of children TreeNodes the receiver contains.

int getIndex(TreeNode node): Returns the index of node in the receivers children.

TreeNode getParent(): Returns the parent TreeNode of the receiver.

Boolean isLeaf(): Returns true if the receiver is a leaf.

Interface MutatableTreeNode

Sub-interface of TreeNode Methods:

void insert(MutableTreeNode child, int index): Adds child to the receiver at index.

void remove(int index): Removes the child at index from the receiver.

void remove(MutableTreeNode node): Removes node from the receiver.

void removeFromParent(): Removes the receiver from its parent.

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

Size = number of nodes in the tree. Recursion analysis:

Base case: empty root => return 0; Recursive case: non-empty root

=> return 1 + sum of size of allchild sub-trees.

Size of a tree

Algorithm size(TreeNode root)Input TreeNode rootOutput: size of the treeif (root = null)

return 0;

elsereturn 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

There are no general tree classes in the standard libraries in many languages: e.g. C++ STL, Ruby, Python, etc. No single interface tree design satisfies all

users. There may be specialized trees, especially

for GUI design. E.g. JTree in Java FXTreeList in Ruby

Tree Traversal

Traversing is the systematic way of accessing, or ‘visiting’ all the nodes in a tree.

Consider the three operations on a binary tree: V: Visit a node L: (Recursively) traverse the left

subtree of a node R: (Recursively) traverse the

right subtree of a node

Tree Traversing

We can traverse a tree in six different orders:

LVRVLR

LRVVRLRVLRLV

Preorder Traversal 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

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Algorithm preOrder(v)visit(v)for each child w of v

preorder (w)

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 In a postorder traversal, a

node is visited after its descendants

Application: compute space used by files in a directory and its subdirectories

Algorithm postOrder(v)for each child w of v

postOrder (w)visit(v)

cs16/

homeworks/todo.txt

1Kprograms/

DDR.java10K

Stocks.java25K

h1c.doc3K

h1nc.doc2K

Robot.java20K

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1

7

2 4 5 6

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Inorder Traversal

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

There are three common notations for expressions: Prefix: operator come before operands. Postfix: operator come after operands. Infix:

Binary operator come between operands. Unary operator come before the operand.

Examples

Infix: (a+b)*(c-d/f/(g+h)) Used by human being.

Prefix: *+ab-c//df+gh E.g. functional language such as Lisp: Lisp: (* (+ a b) (- c (/ d (/ f (+ g h)))))

Postfix: ab+cdf/gh+/-* E.g. Postfix calculator

Expression Trees

An expression tree can be used to represent an expression. Operator: root Operands: child sub-trees Variable and constants: leaf nodes.

Traversals of Expression Trees

Preorder traversal => prefix notation.

Postorder traversal => postfix notation.

Inorder traversal with parenthesis added => infix notation.

Linked Structure for Trees A node is represented

by an object storing Element Parent node Sequence of children

nodes Node objects implement

the Position ADT

B

DA

C E

F

B

A D F

C

E

Tree Classes in Ruby

FXTreeItem: a tree node for GUI application.

Some methods: 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

FXTreeList: a list of FXTreeItem. Include methods for list

manipulation, such as appendItem, clearItems, etc.

Questions and Comments?