2. Basic Tree Concepts
Trees2
A tree consists of finite set of elements,
called nodes, and a finite set of directed
lines called branches, that connect the
nodes.
The number of branches associated
with a node is the degree of the node.
5. Tree
A simple unordered tree; in this diagram, the node
labelled 7 has two children, labelled 2 and 6, and one
parent, labelled 2. The root node, at the top, has no
parent.
A tree is a widely-used data structure that emulates a
hierarchical tree structure with a set of linked nodes.
6. Basic Tree Concepts
Trees6
When the branch is directed toward the
node, it is in degree branch.
When the branch is directed away from
the node, it is an out degree branch.
The sum of the in degree and out
degree branches is the degree of the
node.
If the tree is not empty, the first node is
called the root.
7. Basic Tree Concepts
Trees7
The in degree of the root is, by definition,
zero.
With the exception of the root, all of the nodes
in a tree must have an in degree of exactly
one; that is, they may have only one
predecessor.
All nodes in the tree can have zero, one, or
more branches leaving them; that is, they may
have out degree of zero, one, or more.
9. Basic Tree Concepts
Trees9
A leaf is any node with an out degree of
zero, that is, a node with no successors.
A node that is not a root or a leaf is
known as an internal node.
A node is a parent if it has successor
nodes; that is, if it has out degree greater
than zero.
A node with a predecessor is called a
child.
10. Basic Tree Concepts
Trees10
Two or more nodes with the same parents
are called siblings.
An ancestor is any node in the path from
the root to the node.
A descendant is any node in the path
below the parent node; that is, all nodes
in the paths from a given node to a leaf are
descendants of that node.
11. Basic Tree Concepts
Trees11
A path is a sequence of nodes in which
each node is adjacent to the next node.
The level of a node is its distance from
the root. The root is at level 0, its children
are at level 1, etc. …
12. Basic Tree Concepts
Trees12
The height of the tree is the level of the
leaf in the longest path from the root plus
1. By definition the height of any empty
tree is -1.
A sub tree is any connected structure
below the root.
The first node in the sub tree is known as
the root of the sub tree.
15. Recursive definition of a tree
Trees15
A tree is a set of nodes that either:
is empty or
has a designated node, called the
root, from which hierarchically descend
zero or more sub trees, which are also
trees.
16. Tree Representation
Trees16
General Tree – organization chart
format
Indented list – bill-of-materials system
in which a parts list represents the
assembly structure of an item
19. Parenthetical Listing
Trees19
Parenthetical Listing – the algebraic
expression, where each open
parenthesis indicates the start of a new
level and each closing parenthesis
completes the current level and moves
up one level in the tree.
21. Trees21
Binary Trees
A binary tree can have no more than twoA binary tree can have no more than two
descendentsdescendents
• Properties
• Binary Tree Traversals
• Expression Trees
• Huffman Code
22. Binary Trees
Trees22
A binary tree is a tree in which no node
can have more than two sub trees; the
maximum out degree for a node is two.
In other words, a node can have zero,
one, or two sub trees.
These sub trees are designated as the
left sub tree and the right sub tree.
25. Some Properties of Binary Trees
The height of binary trees can be mathematically
predicted
Given that we need to store N nodes in a binary
tree, the maximum height is
maxH N=
Trees25
A tree with a maximum height is rare. It
occurs when all of the nodes in the entire
tree have only one successor.
26. Some Properties of Binary Trees
The minimum height of a binary tree is determined
as follows:
[ ]min 2log 1H N= +
Trees26
For instance, if there are three nodes to be
stored in the binary tree (N=3) then Hmin=2.
27. Some Properties of Binary Trees
Given a height of the binary tree, H, the
minimum number of nodes in the tree is
given as follows:
minN H=
Trees27
28. Some Properties of Binary Trees
The formula for the maximum number of nodes is
derived from the fact that each node can have
only two descendents.
Given a height of the binary tree, H, the
maximum number of nodes in the tree is given
as follows:
max 2 1H
N = −
Trees28
29. Some Properties of Binary Trees
The children of any node in a tree can be accessed by
following only one branch path, the one that leads to the
desired node.
The nodes at level 1, which are children of the root, can be
accessed by following only one branch; the nodes of level
2 of a tree can be accessed by following only two branches
from the root, etc.
The balance factor of a binary tree is the difference in
height between its left and right sub trees:
L RB H H= −Trees29
31. Some Properties of Binary Trees
In the balanced binary tree (the height of its sub trees
differs by no more than one) its balance factor is -1, 0, or
1), and its sub trees are also balanced.
Trees31
32. Complete and nearly complete binary
trees
Trees32
A complete tree has the maximum
number of entries for its height. The
maximum number is reached when the last
level is full.
A tree is considered nearly complete if it
has the minimum height for its nodes and
all nodes in the last level are found on the
left
36. Traversal through BSTs
Remember that a binary tree is either empty or
it is in the form of a Root with two sub trees.
If the Root is empty, then the traversal
algorithm should take no action (i.e., this is
an empty tree -- a "degenerate" case).
If the Root is not empty, then we need to
print the information in the root node and start
traversing the left and right sub trees.
When a sub tree is empty, then we stop
traversing it.
37. Traversal through BSTs
The recursive traversal algorithm is:
Traverse (Root)
If the Tree is not empty then
Visit the node at the Root
Traverse(Left sub tree)
Traverse(Right sub tree)
When traversing any binary tree, the algorithm should
have 3 choices of when to process the root:
before it traverses both sub trees,
after it traverses the left sub tree,
or after it traverses both sub trees.
Each of these traversal methods has a name: preorder, in
order, post order
38. Binary Tree Traversal
Trees38
A binary tree traversal requires that each
node of the tree be processed once and
only once in a predetermined sequence.
In the depth-first traversal processing
process along a path from the root through
one child to the most distant descendant of
that first child before processing a second
child.
47. …In order traversal
It would traverse the same tree as:
5,10,15,20,30,40,60,65,70,85;
Notice that this type of traversal produces the numbers in
order.
Search trees can be set up so that all of the nodes in the
left sub tree are less than the nodes in the right sub tree.
60
20
70
10
5 15 30
40 65 85
51. Tree Representations
There are many different ways to represent trees.
Common representations represent the nodes as records
allocated on the heap with pointers to their children, their
parents, or both, or as items in an array, with
relationships between them determined by their positions
in the array (e.g., binary heap).
52. Trees and Graphs
The tree data structure can be generalized to represent
directed graphs by removing the constraints that a node
may have at most one parent, and that no cycles are
allowed.
Edges are still abstractly considered as pairs of nodes,
however, the terms parent and child are usually replaced
by different terminology (for example, source and target).
53. Relationship with Trees in Graph Theory
• In graph theory, a tree is a connected acyclic graph; unless
stated otherwise, trees and graphs are undirected.
• There is no one-to-one correspondence between such
trees and trees as data structure.
• We can take an arbitrary undirected tree, arbitrarily pick
one of its vertices as the root, make all its edges directed
by making them point away from the root node - producing
an arborescence and assign an order to all the nodes.
The result corresponds to a tree data structure.
Picking a different root or different ordering produces a
different one.
54. • Enumerating all the items
• Enumerating a section of a tree
• Searching for an item
• Adding a new item at a certain position on the tree
• Deleting an item
• Removing a whole section of a tree is called pruning
• Adding a whole section to a tree is called grafting
• Finding the root for any node
Common Operations
55. Common Uses
Manipulate hierarchical data
Make information easy to search
Manipulate sorted lists of data
As a workflow for composting digital images for
visual effects
Router algorithms
56. Search Trees
A binary search tree can be created so that the
elements in it satisfy an ordering property.
This allows elements to be searched for quickly.
All of the elements in the left sub-tree are less than the
element at the root which is less than all of the
elements in the right sub-tree and this property applies
recursively to all the sub-trees.
The great advantage of this is that when searching for
an element, a comparison with the root will either find
the element or indicate which one sub-tree to search.
The ordering is an invariant property of the search
tree. All routines that operate on the tree can make
use of it provided that they also keep it holding true.
57. Additional…Binary Search Trees
Key property
Value at node
Smaller values in left sub-tree
Larger values in right sub-tree
Example
X > Y
X < Z
Y
X
Z
59. Example Binary Searches
Find ( root, 2 )
5
10
30
2 25 45
5
10
30
2
25
45
10 > 2, left
5 > 2, left
2 = 2, found
5 > 2, left
2 = 2, found
root
60. Example Binary Searches
Find (root, 25 )
5
10
30
2 25 45
5
10
30
2
25
45
10 < 25, right
30 > 25, left
25 = 25, found
5 < 25, right
45 > 25, left
30 > 25, left
10 < 25, right
25 = 25, found
61. Types of Binary Trees
Degenerate – only one child
Complete – always two children
Balanced – “mostly” two children
more formal definitions exist, above are intuitive ideas
Degenerate
binary tree
Balanced
binary tree
Complete
binary tree
62. Binary Trees Properties
Degenerate
Height = O(n) for n
nodes
Similar to linked list
Balanced
Height = O( log(n) ) for
n nodes
Useful for searches
Degenerate
binary tree
Balanced
binary tree
63. Binary Search Properties
Time of search
Proportional to height of tree
Balanced binary tree
O( log(n) ) time
Degenerate tree
O( n ) time
Like searching linked list / unsorted
array
64. Binary Search Tree Construction
How to build & maintain binary trees?
Insertion
Deletion
Maintain key property (invariant)
Smaller values in left sub tree
Larger values in right sub tree
65. Example Insertion
Insert ( 20 )
5
10
30
2 25 45
10 < 20, right
30 > 20, left
25 > 20, left
Insert 20 on left
20
67. Example Deletion (Internal Node)
Delete ( 10 )
5
10
30
2 25 45
5
5
30
2 25 45
2
5
30
2 25 45
Replacing 10
with largest
value in left
subtree
Replacing 5
with largest
value in left
subtree
Deleting leaf
68. Example Deletion (Internal Node)
Delete ( 10 )
5
10
30
2 25 45
5
25
30
2 25 45
5
25
30
2 45
Replacing 10
with smallest
value in right
subtree
Deleting leaf Resulting tree
69. Balanced Search Trees
Kinds of balanced binary search trees
height balanced vs. weight balanced
“Tree rotations” used to maintain balance on
insert/delete
Non-binary search trees
2/3 trees
each internal node has 2 or 3 children
all leaves at same depth (height balanced)
71. AVL Tree
AVL (Adelson-Velskii and Landis) tree.
Also called Height Balanced Binary Search Trees
An AVL tree is identical to a BST except
Height of the left and right sub trees can differ
by at most 1.
Height of an empty tree is defined to be (–1).
Every sub tree is an AVL tree.
75. An example of an AVL tree where the
heights are shown next to the nodes
88
44
17 78
32 50
48 62
2
4
1
1
2
3
1
1
76. Balanced Binary Tree
The height of a binary tree is the maximum level of
its leaves (also called the depth).
The balance of a node in a binary tree is defined as
the height of its left sub tree minus height of its right
sub tree.
Each node has an indicated balance of 1, 0, or –1.
77. B-Tree
A B-Tree of order m is an m-way tree, such that:
All leaves are on the same level
All internal nodes except the root are constrained to have
at most non empty children and at least m/2 non empty
children
The root has at most m non empty children
78. B-Tree of order 2, also known as 2-3-4-tree:
1717 2121
77
11
11
11
88
22
00
22
66
33
11
22 44 55 66 88 99
11
22
11
66
2222 2323 2525 2727 2929 3030 3232 3535
79. B- Tree
A B-tree is a tree data structure that keeps data sorted
and allows searches, sequential access, insertions, and
deletions in logarithmic time.
The B-tree is a generalization of a binary search tree in
that a node can have more than two children. B-tree is
optimized for systems that read and write large blocks of
data.
It is commonly used in databases and file systems.
80. ...B- Tree
A B-tree is a specialized multi-way tree designed
especially for use on disk. In a B-tree each node may
contain a large number of keys.
The number of sub trees of each node, then, may also be
large.
A B-tree is designed to branch out in this large number of
directions and to contain a lot of keys in each node so
that the height of the tree is relatively small.
This means that only a small number of nodes must be
read from disk to retrieve an item.
The goal is to get fast access to the data, and with disk
drives this means reading a very small number of
records.
81. B - Tree
For example, the following is a multiway search tree of
order 4. Note that the first row in each node shows the
keys, while the second row shows the pointers to the
child nodes.
There is a record of data associated with each key, so
that the first row in each node might be an array of
records where each record contains a key and its
associated data.
Another approach would be to have the first row of each
node contain an array of records where each record
contains a key and a record number for the associated
data record, which is found in another file.
This is often used when the data records are large.
83. ....
Records are stored in locations called leaves.
This name derives from the fact that records always exist
at end points; there is nothing beyond them.
The maximum number of children per node is the order of
the tree.
The number of required disk accesses is the depth.
The image at left shows a binary tree for locating a
particular record in a set of eight leaves.
84. The image at right shows a B-tree of order three for
locating a particular record in a set of eight leaves (the
ninth leaf is unoccupied, and is called a null).
The binary tree at left has a depth of four; the B-tree at
right has a depth of three.
Clearly, the B-tree allows a desired record to be located
faster, assuming all other system parameters are identical.
A sophisticated program is required to execute the
operations in a B-tree. But this program is stored in RAM,
so it runs fast.
