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Chapter 9
Binary Tree and
General Tree
Overview
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Two-way decision making is one of the
fundamental concepts in computing.
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A binary tree models two-way decisions.
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A hierarchy represents multi-way choices.
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The general tree is an extension of the binary tree.
Learning Objectives
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Describe a binary tree in terms of its structure
and components, and learn recursive definitions
of the binary tree and its properties.
Study standard tree traversals in depth.
Develop a binary tree class interface based on
its recursive definition.
Learn about the signature of a binary tree and
understand how to build a binary tree given its
signature.
Learning Objectives
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Understand Huffman coding, a binary treebased text compression application, and use
the binary tree class to implement Huffman
coding.
Implement the binary tree class.
Study how tree traversals may be implemented
non-recursively using a stack.
Describe the properties of a general tree.
Learning Objectives
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Learn the natural correspondence of a general
tree with an equivalent binary tree, and the
signature of a general tree.
9.1.1 Components
9.1.1 Components
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A binary tree consists of nodes and branches.
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A node is a place in the tree where data is stored.
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There is a special node called the root.
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The nodes are connected to each other by links or
branches.
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“starting point” of the tree.
A left branch or right branch.
Binary means that there are at most two choices.
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A node is said to have at most two children.
A node that does not have any children is called a leaf.
Non-leaf nodes are called internal nodes.
9.1.1 Components
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There is a single path from any node to any
other node in the tree.
9.1.2 Position as Meaning
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If-then-else tree
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Represents an if-thenelse construct in a
program.
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Every node in this tree
is conditional
expression that
evaluates to yes or no.
If it evaluates to yes,
the left branch (if any)
is taken and if
evaluates to no, the
right branch (if any) is
taken.
9.1.2 Position as Meaning
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Expression tree
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(f + ((a * b) - c))
Double-click to add graphics
Create Expression tree
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1- If the current token is a '(', add a new node as the
left child of the current node, and descend to the left
child.
2- If the current token is in the list ['+','-','/','*'], set the
root value of the current node to the operator
represented by the current token. Add a new node
as the right child of the current node and descend to
the right child.
3- If the current token is a number, set the root
value of the current node to the number and return
to the parent.
4- If the current token is a ')', go to the parent of the
current node.
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let’s look at an example of the rules outlined
above in action. We will use the
expression (3+(4∗5)).
We will parse this expression into the following
list of character tokens['(', '3', '+', '(', '4', '*', '5'
,')',')'].
Initially we will start out with a parse tree that
consists of root node.
(3+(4∗5))
9.1.3 Structure
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Structure
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Two trees with the same number of nodes may not
have the same structure.
9.1.3 Structure
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Depth is the distance
from the root.
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Nodes at the same
depth are said to be at
the same level, with
the root being at level
zero.
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The height of a tree is
the maximum level (or
depth) at which there
is a node.
9.1.3 Structure
• Full Binary Tree: A binary tree in which all of the
leaves are on the same level and every nonleaf node
has two children
9.1.3 Structure
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(a), first three are strictly binary, but the fourth is
not. first two are FULL binary tree
(b), first two are complete and the last two are
not.
At level i, there can be at most 2i nodes.
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Maximum number of nodes over all the levels.
9.1.4 Recursive Definitions
9.1.4 Recursive Definitions
Traversal Definitions
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Preorder traversal: Visit the root, visit the left
subtree, visit the right subtree
Inorder traversal: Visit the left subtree, visit the
root, visit the right subtree
Postorder traversal: Visit the left subtree, visit
the right subtree, visit the root
Visualizing Binary Tree Traversals
Three Binary Tree Traversals
9.2 Binary Tree Traversals
9.2 Binary Tree Traversals
Binary Search Tree
Overview
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A binary tree possesses ordering property that
maintains the data in its nodes in sorted order.
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Since the search tree is a linked structure, entries
may be inserted and deleted without having to
move other entries over, unlike ordered lists in
which insertions and deletions require data
movement.
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The AVL tree is a height-balanced binary search
tree that delivers guaranteed worst-case search,
insert, and delete times that are all O(log n)
Learning Objectives
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Explore the motivation for binary search trees
by learning about the comparison tree for binary
search.
Use the comparison tree as an analytical tool to
determine the running time of binary search.
Describe the binary search tree structure and
properties.
Study the primary binary search tree operations
of search, insert, and delete, and analyze their
running times.
Learning Objectives
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Understand a binary search tree class interface
and use it in application examples.
Implement the binary search tree class with a
binary tree class as the reused storage
component.
Study the AVL tree structure properties, the
search, insert, and delete operations, and their
running times.
10.2 Binary Search Tree Properties
10.2 Binary Search Tree Properties
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All three trees have the same set of keys.
Their structures are different, depending on the
sequence of insertion or deletion.
10.3 Binary Search Tree Operations
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Three foundational operations
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Search
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Insert
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Delete
10.3.1 Search
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The tree nodes are for real.
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The target key is compared against the key at the root of
the tree.
If they are equal, sucess.
If not, recusively search the appropriate child.
Search terminates with failure if an empty subtree is
reached.
10.3.1 Search
10.3.2 Insert
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To insert a value, search must force a failure.
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Item in inserted in the failed location.
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A newly inserted node always becomes a leaf
node in the search tree.
10.3.2 Insert
10.3.3 Delete
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The value to be deleted is first located in the
binary search tree.
Three possible cases.
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Case a: X is a leaf node.
10.3.3 Delete
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Case b: X has one child
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Replace the deleted node with the child.
10.3.3 Delete
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Case c: X has two children
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Find the inorder predecessor, Y, of X.
Copy the entry at Y into X.
Apply deletion on Y.
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Applying deletion on Y will revert to either case b or a since Y is
guaranteed to not have a right subtree.
Running Times
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Search: worst case
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Tied to the worst possible shape a tree can attain.
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Such a tree degenerates into sequential search.
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O(n).
Running Times
Running Times
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Insertion: worst case
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O(n)
Deletion: worst case
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O(n)
Balancing
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Keeping a binary search tree balanced allows
the height never to exceed O(log n).
There are two popular ways of maintaining and
constructing balanced binary search trees.
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AVL tree.
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red-black tree.
10.4 A BinarySearchTree Class
10.4 A BinarySearchTree Class
10.5.1 Example Treesort
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inOrder traversal method invokes visitor.visit()
when a node is visited.
10.5.1 Example: Treesort
10.5.2 Example: Counting Keys
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Count the number of keys in a binary search
tree that are less than a given key.
It is possible to simply examine every node of
the tree.
10.6 BinarySearchTree Class
Implementation
10.6 BinarySearchTree Class
Implementation
10.6.1 Search Implementation
10.6.2 Insert Implementation
10.6.2 Insert Implementation
10.6.2 Insert Implementation
10.6.2 Insert Implementation
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left, right, and parent fields are directly
accessed, instead of calling attachLeft and
attachRight.
Clients of BinaryTree that are not in its package
are required to use attachLeft or attachRight.
10.6.3 Delete Implementation
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findPredecessor helper method.
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Case c requires finding the inorder predeccessor.
10.6.3 Delete Implementation
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deleteHere helper method.
10.6.3 Delete Implementation
10.6.3 Delete Implementation
10.6.3 Delete Implementation
10.6.4 Convenience Methods and
Traversals
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minValue needs to find the "leftmost" node.
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maxValue finds the "rightmost".
10.6.4 Convenience Methods and
Traversals
10.6.4 Convenience Methods and
Traversals
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This implementation hides the tree structure
from its clients.
All clients need to see are the search, insert,
and delete operation.
"preorder", "inorder", and "postorder" provide a
small window into the implementation structure.
10.9 Summary
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A comparison tree for binary search on an array
is a binary tree that depicts all possible search
paths.
A failure node in a comparison tree catches a
range of values that lie between its inorder
predecessor and its in order successor in the
tree.
A comparison tree is simply a conceptual tool to
analyze the time taken by binary search, and is
therefore often referred to as an implicit search
tree.
10.9 Summary
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The shape of the comparison tree is
independent of the data entries in the array; it
only depends on the length of the array.
The worst-case number of comparisons for a
successful search in 2h-1, and for unsuccessful
search is 2h.
Binary search on an ordered array is O(log n).
A binary search tree is a binary tree whose
entries are arranged in order.
10.9 Summary
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An inorder traversal of a binary search tree will
visit the nodes in ascending order of values.
The values stored in a binary search tree must
lend themselves to being arranged in order.
For any given number of values, n, there is only
one binary search comparison tree. There are
many binary search trees possible as there are
different binary trees that can be constructed
out of n nodes.
10.9 Summary
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The worst possible binary tree structure is one
that is completely skewed either to the left or
right – O(n).
The worst-case running times for search, insert,
and delete in a balanced binary search tree are
all O(log n).
Treesort is an algorithm to sort a set of values
by inserting them one by one into a binary
search tree, and then visiting them in inorder
sequence -- O(n2).
10.9 Summary
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The AVL tree and red-black tree are two of the
many types of balanced binary search trees that
guarantee a worst case search / insert / delete
time of O(log n).
An AVL tree is a binary search tree in which the
heights of the left and right subtrees of every
node differ by at most 1.
Recursive definition: An AVL tree is a binary
search tree in which the left and right subtrees
of the root are AVL trees whose heights differ by
at most 1.
10.9 Summary
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Rotation about a link in an AVL tree takes O(1)
time.
Insertion in an AVL tree starts with a regular
binary search tree insertion, followed by
rebalancing.
Deletion in an AVL tree starts with a regular
binary search tree deletion, followed by
rebalancing.
9.3 A Binary Tree Class
9.3 A Binary Tree Class
9.3 A Binary Tree Class
9.3 A Binary Tree Class
9.3 A Binary Tree Class
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Delete a single node from a tree.
9.3 A Binary Tree Class
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Recursive traversal procedures.
9.3 A Binary Tree Class
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Running times of methods
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makeRoot, setData, and getData involve reading or
writing data once.
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Checking for whether the pointer is null.
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isEmpty O(1).
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clear simply have to set the root pointer to null.
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O(1)
attachLeft and attachRight
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Three pointer settings O(1)
detachLeft and detachRight O(1)
9.3 A Binary Tree Class
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Root called on a leaf node that is at the greatest
possible depth in the tree.
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O(h) h is the height of the tree.
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The height could be as much as n – 1 in the case
where the tree is entirely lopsided.
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O(n)
9.9 Summary
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Binary trees model two-way decision making
systems.
A binary tree consists of nodes and branches.
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There is a special node called the root.
Every node in a binary tree has at most two
children.
Nodes that have no children are called leaf
nodes, others are called internal nodes.
9.9 Summary
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There is a single path between any pair of
nodes in a binary tree.
A binary tree is defined by the relative positions
of the data in its nodes, and the tree as a whole
carries a meaning that would change if the
relative positions of the data in the tree were to
change.
The depth of a node in a binary tree is the
number of branches (distance) from the root to
that node.
9.9 Summary
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Nodes at the same depth in a binary level are
said to be at the same level.
The height of a binary tree is the maximum level
at which there is a node.
A strictly binary tree is one in which every node
has either no child or two children.
In a complete binary tree, every level but the
last must have the maximum number of nodes
possible at that level.
9.9 Summary
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The maximum possible number of nodes at
level i in a binary tree is 2i.
The maximum possible number of nodes in a
binary tree of height h is 2h+1 – 1.
If Nmax is the maximum number of nodes in a
binary tree, its height is log(Nmax + 1) – 1.
Recursive definition: A binary tree is either
empty, or it consists of a special node called
root that has a left subtree and a right subtree
that are mutually disjoint binary trees.
9.9 Summary
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The number of nodes in an empty binary tree is
zero.
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Otherwise, the number of nodes is one plus the
number of nodes each in the left and right subtrees
of the root.
The height of an empty binary tree is -1.
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Otherwise, the height is one plus the maximum of
the heights of the left and right subtrees of the root.
9.9 Summary
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Recursive definition of inorder traversal type T:
first recursively traverse the Left subtree of T,
then Visit the root of T, then recursively traverse
the Right subtree of T.
Recursive definition of preorder traversal of tree
T: first Visit the root of T, then recursively
traverse the Left subtree of T, then recursively
traverse the Right subtree of T.
9.9 Summary
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Recursive definition of postorder traversal of
treeT: first recursively traverse the Left subtree
of T, then recursively traverse the Right subtree
of T, then Visit the root of T.
The recursive traversals may be written in short
form as follows: inorder is LVR, preorder is
VLR, and postorder is LRV.
Level-order traversal of treeT: starting at the
root level, go level by level in T, visiting the
nodes at any level in left to right order.