Transcript Trees

Trees
Definition of a tree
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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
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Like a binary tree, a tree has a
root, internal nodes, and leaves
A
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Each node contains an element
and has branches leading to other
nodes (its children)
B
C
D
E
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Each node (other than the root)
has a parent
F G H
I
J
K
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Each node has a depth (distance
from the root)
L
M
N
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More definitions
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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
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Data structure for a tree
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A node in a binary tree can be represented as follows:
class BinaryTreeNode {
Object value;
BinaryTreeNode leftChild, rightChild;
}
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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 an ArrayList
class TreeNode {
Object value;
ArrayList children;
}
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If we don’t care about the order of children, we might use a Set
instead of a ArrayList
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ADT for a tree
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It must be possible to:
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Construct a new tree
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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:
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Remove a child (and the subtree rooted at that child)
Get the parent of a node
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Traversing a tree
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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());
}
}
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You can traverse a tree in postorder:
void postorderPrint(node) {
Iterator iter = node.children.iterator();
while (iter.hasNext()) {
postorderPrint(iter.next());
}
System.out.println(node);
}
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You can’t usually traverse a tree in inorder
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Why not?
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Other tree manipulations
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There’s really nothing new to talk about; you’ve
seen it all with binary trees
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A tree consists of nodes, each node has references
to some other nodes—you know how to do all this
stuff
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There are some useful algorithms for searching
trees, and with some modifications they also apply
to searching graphs
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Let’s move on to some applications of trees
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File systems
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File systems are almost always implemented as a tree
structure
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The nodes in the tree are of (at least) two types: folders (or
directories), and plain files
A folder typically has children—subfolders and plain files
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A folder also contains a link to its parent—in 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
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Family trees
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It turns out that a tree is not a good way to
represent a family tree
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Every child has two parents, a mother and a father
Parents frequently remarry
An “upside down” binary tree almost works
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Since it is a biological fact (so far) that every child has
exactly two parents, we can use left child = father and
right child = mother
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
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Part of a genealogy
Chester
Elaine
Eugene
Pauline
Paul
a
David
Steven
Winfred
Carol
Danielle
Isaac
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Game trees
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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
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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—
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You can’t 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
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Binary trees for expressions
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Ordered trees can be used to represent arithmetic expressions
+
2
+
2
The expression 2+2
2
*
+
*
3
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The expression 2+3*4
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2
4
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The expression (2+3)*4
To evaluate an expression (given as a node):
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If it is a leaf, the element in it specifies the value
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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, the element in it specifies an operation
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Evaluate the children and combine them according to the operation
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(General) trees for expressions
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You can use binary trees for expressions if you have only unary
and binary operators
Java has a ternary operator
if
?:
>
x
>
x y
y
The expression x > y ? x : y
x
=
=
y max x max y
The statement if (x > y) max = x;
else max = y;
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Trees can be used to represent statements as well as
expressions
Statements can be evaluated as easily as expressions
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More trees for statements
while (n >= 1) {
exp = x * exp;
n--;
}
for (int i = 0; i < n; i++)
a[i] = 0;
for
while
>=
n
=
;
1
=
exp
x
*
--
int
n
i
<
0 i
n
++
=
i
[] 0
a
i
exp
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Writing compilers and interpreters
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A compiler does three things:
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An interpreter does three things:
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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
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 don’t
care too much about efficiency or good error messages
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I’ll never need to write a compiler...
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Are you sure?
If you can’t 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:
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tell Mary "Meet me at noon"
fire phasers at 3, 7
17.25, 0.203 + 8.97i, 0.95i
28°12"48'
3:30pm-5pm
Parsers are built on top of tokenizers
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Java provides basic support for tokenizing with its StringTokenizer and
StreamTokenizer classes
Often, however, you need to use these to build a more specific tokenizer
Usually, the best way to build a tokenizer is as a state machine
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The End
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