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Trees

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An empty tree has no nodes ... However, each node in a tree has an arbitrary number of children, so we need ... public interface Tree ... – PowerPoint PPT presentation

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Title: Trees


1
Trees
2
Definition of a tree
  • A tree is like a binary tree, except that a node
    may have any number of children
  • Depending on the needs of the program, the
    children may or may not be ordered
  • Like a binary tree, a tree has a root, internal
    nodes, and leaves
  • Each node contains an element and has branches
    leading to other nodes (its children)
  • Each node (other than the root) has a parent
  • Each node has a depth (distance from the root)

3
More definitions
  • An empty tree has no nodes
  • The descendents of a node are its children and
    the descendents of its children
  • The ancestors of a node are its parent (if any)
    and the ancestors of its parent
  • The subtree rooted at a node consists of the
    given node and all its descendents
  • An ordered tree is one in which the order of the
    children is important an unordered tree is one
    in which the children of a node can be thought of
    as a set
  • The branching factor of a node is the number of
    children it has
  • The branching factor of a tree is the average
    branching factor of its nodes

4
Data structure for a tree
  • A node in a binary tree can be represented as
    follows
  • class BinaryTreeNode Object value
    BinaryTreeNode leftChild, rightChild
  • However, each node in a tree has an arbitrary
    number of children, so we need something that
    will hold an arbitrary number of nodes, such as a
    Vector or a linked list
  • class TreeNode Object element Vector
    children
  • If we dont care about the order of children, we
    might use a Set instead of a Vector

5
ADT for a tree
  • It must be possible to
  • Construct a new tree
  • If a tree can be empty, this may require a header
    node
  • Add a child to a node
  • Get (iterate through) the children of a node
  • Access (get and set) the value in a node
  • It should probably be possible to
  • Remove a child (and the subtree rooted at that
    child)
  • Get the parent of a node

6
A Tree ADT, I
  • Here is a Tree ADT defined in Java Collections by
    David A. Watt and Deryck F. Brown
  • public interface Tree // ...method
    declarations... public interface Node
    public Object getElement() public void
    setElement(Object elem)
  • An interesting aspect of this ADT is that it
    uses an inner interface
  • An interface cant have an inner class
  • The details are not important for our purposes
  • The inner interface is referred to by Tree.Node

7
A Tree ADT, II
  • public interface Tree
  • // Accessors
  • public Tree.Node root()
  • public Tree.Node parent(Tree.Node node)
  • public int childCount(Tree.Node node)
  • // Transformers
  • public void makeRoot(Object elem)
  • public Tree.Node addChild(Tree.Node node,
    Object elem)
  • public void remove(Tree.Node node)
  • // Iterator
  • public Iterator children(Tree.Node node)
  • // Inner interface for tree nodes
  • public interface Node public Object
    getElement() public void
    setElement(Object elem)

8
Traversing a tree
  • You can traverse a tree in preorder
  • void preorderPrint(node) System.out.println(
    node) Iterator iter node.children.iterator(
    ) while (iter.hasNext())
    preorderPrint(iter.next())
  • You can traverse a tree in postorder
  • void postorderPrint(node) Iterator iter
    node.children.iterator() while
    (iter.hasNext()) postorderPrint(iter.nex
    t()) System.out.println(node)
  • You cant usually traverse a tree in inorder
  • Why not?

9
Other tree manipulations
  • Theres really nothing new to talk about youve
    seen it all with binary trees
  • A tree consists of nodes, each node has
    references to some other nodesyou know how to do
    all this stuff
  • There are some useful algorithms for searching
    trees, and with some modifications they also
    apply to searching graphs
  • Lets move on to some applications of trees

10
File systems
  • File systems are almost always implemented as a
    tree structure
  • The nodes in the tree are of (at least) two
    types folders (or directories), and plain files
  • A folder typically has childrensubfolders and
    plain files
  • A folder also contains a link to its parentin
    both Windows and UNIX, this link is denoted by ..
  • In UNIX, the root of the tree is denoted by /
  • A plain file is typically a leaf

11
Family trees
  • It turns out that a tree is not a good way to
    represent a family tree
  • Every child has two parents, a mother and a
    father
  • Parents frequently remarry
  • An upside down binary tree almost works
  • Since it is a biological fact (so far) that every
    child has exactly two parents, we can use left
    child mother and right child father
  • The terminology gets a bit confusing
  • If you could go back far enough, it becomes a
    mathematical certainty that the mother and father
    have some ancestors in common

12
Part of a genealogy
13
Game trees
  • Trees are used heavily in implementing games,
    particularly board games
  • A node represents a position on the board
  • The children of a node represent all the possible
    moves from that position
  • More precisely, the branches from a node
    represent the possible moves the children
    represent the new positions
  • Planning ahead (in a game) means choosing a path
    through the tree
  • However
  • You cant have a cycle in a tree
  • If you can return to a previous position in a
    game, you have a cycle
  • Graphs can have cycles

14
Binary trees for expressions
  • Ordered trees can be used to represent arithmetic
    expressions
  • To evaluate an expression (given as a node)
  • If it is a leaf, the element in it specifies the
    value
  • If the element is a number, that number is the
    value
  • If the element is a variable, look up its value
    in a table
  • If it is not a leaf,
  • Evaluate the children and combine them according
    to the operation specified by the element

15
(General) trees for expressions
  • You can use binary trees for expressions if you
    have only unary and binary operators
  • Java has a ternary operator
  • Trees can be used to represent statements as well
    as expressions
  • Statements can be evaluated as easily as
    expressions

16
More trees for statements
  • while (n 1) exp x exp n--
  • for (int i 0 i

17
Writing compilers and interpreters
  • A compiler does three things
  • Parses the input program (converts it into an
    abstract syntax tree)
  • (Optionally) optimizes the abstract syntax tree
  • Traverses the tree and outputs assembly language
    or machine code to do the same operations
  • An interpreter does three things
  • Parses the input program (converts it into an
    abstract syntax tree)
  • (Optionally) optimizes the abstract syntax tree
  • Traverses the tree in an order controlled by the
    node contents, and performs the operations as it
    goes
  • Parsing is usually the hard part, but there is a
    very simple technique (called recursive descent
    parsing) that can be used if the language is
    carefully designed and you dont care too much
    about efficiency or good error messages

18
Ill never need to write a compiler...
  • Are you sure?
  • If you cant parse text inputs, you are limited
    to reading simple things like numbers and Strings
  • If you can parse text input, you can make sense
    of
  • tell Mary "Meet me at noon"
  • fire phasers at 3, 7
  • 17.25, 0.203 8.97i, 0.95i
  • 2812"48'
  • 330pm-5pm
  • Parsing is less important in these days of GUIs,
    but its still pretty important
  • Java provides basic support for parsing with its
    StringTokenizer and StreamTokenizer classes

19
The End
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