datastructures

1. Lecture 4
Dynamic Data Structure Part 2
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2. Content Lecture 4 Dynamic Data
Part 2
 Tree structures
 General
 Tree Representation
 Balanced Trees
 Binary Tree
 Sorted Binary Tree
 Binary Search Tree
 Insert an element in a Sorted Binary Tree
 Delete a node in a Sorted Binary Tree
 Other Tree Types
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3. Tree structures
General
 Trees are one of the most important data structures
 There are several different variants
 We will have a closer look to some of them
 Trees are used in computer science for data storage,
searching and sorting
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4. Tree structures
Definition
A Tree is a finite amount T of one or several nodes such that
 There is a significant node called root(T)
 The other nodes can be divided in disjunctive amount
T1, .. Tm
 Each of these amounts is again a Tree
 These Trees are called Sub Trees
 This is a recursive definition
 At the end a Tree with only one node remains
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5. Tree Representation
 You can display Trees in different ways
 In this course outline Trees are illustrated with the root on
the top and the Sub Trees below
 This is a common illustration
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FED
B C
A

6. Tree Representation
 These are some different representations
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7. Tree Representation
Family Trees
 A common example for trees are Family Trees
 Two variants of Family Trees exist:
 Starting from an ancestor as root and illustrated all the
descendants
 Starting from a descendant as root and illustrated all the
ancestors
 In Family Trees you can have redundancies if some of the
ancestors have the same ancestors
 In this case the entries represent the role of the ancestor
(for example grandmother on the mother’s side)
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10. Tree structures
Definitions for Trees
 Each node in a Tree is the root of one of the Sub Trees
 Each Tree has zero or more child nodes which are below in
the Tree
 A node with a child node is called parent node to the child
(also ancestor node or superior)
 A root node is a node with no parents
 Each Tree has at least one root node
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11. Tree structures
Example
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G
FED
B C
A root node
parent node of D, E, Fchild
node of A

12. Tree structures
 The depth of a node is the length of the path to its root
 A node p is an ancestor to node q if p exists on the path
from q to root
 The node q is called a descendant of p
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13. Tree structures
Example
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G
FED
B C
A
depth of F is 2
C is ancestor of D
D is a descendant of C

14. Tree structures
 The size of a node is the number of descendants a node has
including itself
 Siblings are nodes that share the same parent node
 Nodes without a child node are called leaf node or
terminal nodes
 Internal or inner node are nodes with one or more child
nodes therefore size > 0
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15. Tree structures
Example
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G
FED
B C
A size of A is 7
B and C are siblings
G is a leave node
F is an inner node

16. Tree structures
 In-degree of a node is the number of edges arriving at that
node
 The only node with In-degree = 0 is the root node
 Out-degree of a node is the number of edges leaving that
node
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17. Tree structures
Example
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G
FED
B C
A
In-degree of C 1
Out-degree of C is 3
In-degree of D 1
Out-degree
of D is 0

18. Tree structures
 The level of a node is defined
as:
 The level of the root(T) is 0
 The level of any other node
is increased by 1 to the
level of the node which is
the root of the superior sub
tree
level(node) = level(parent) + 1
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G
FED
B C
A Level 0
Level 1
Level 2
Level 3

19. Balanced Trees
 If the relative order of the Sub Trees is important you call it
a Balanced Tree
 If the order is not important this is called an Oriented
Tree
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G
FED
B C
A
G
EFD
B C
A
These Trees are
different if you look at
them as Balanced Tree
but they are consider
as the same if you look
at them as Oriented
Trees

20. Binary Trees
 A Binary Tree is a finite amount of nodes where the
amount is empty or contains a root and two disjunctive
Binary Trees
 These two Binary Trees are called left and right Sub Tree
of the root
 That means that every node of the tree has zero, one or two
child nodes
 Binary Trees are not a special case of Trees in general
 For example you can have an empty amount as a Binary
Tree but not as a General Tree
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21. Binary Trees
 Any data in the Tree structure can be reached by starting at
the rood node and following either the left or the right
child
 Binary Trees are used for implementation of Binary Tree
Search and Binary Heaps
 Examples for Binary Trees are the elimination contest in
tennis or other sport competitions
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22. Binary Trees
Example
 These two Binary Trees are not the same
 One has left and the other a right Sub Tree
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B
A
B
A
Binary tree with a
left sub tree
Binary tree with a
right sub tree

23. Binary Trees
Implementation
 Binary Trees are easily implemented as a data type like
 One pointer points to the root
 If this pointer is null than the Tree is empty
 A node in the Binary Tree contains a pointer to an object
or a record and the two pointers to the left and the right
Sub Tree
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24. Binary Trees
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root
A
B
C
E FD
G H J
ǁ
ǁ
ǁ ǁ ǁ
ǁ
ǁ
ǁ ǁ ǁ

25. Binary Trees
 There are three possibilities to traverse a Binary Tree:
preorder, inorder and postorder
 If the Tree is empty there is nothing to do
 Otherwise the traverse possibilities are defined as:
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preorder visit the root
traverse the left sub tree
traverse the right sub tree
inorder traverse the left sub tree
visit the root
traverse the right sub tree
postorder traverse the left sub tree
traverse the right sub tree
visit the root
B C
A
A B C
B A C
B C A

26. Binary Trees
Examples
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preorder: A B D C E G F H I
inorder: D B A E G C H F I
postorder: D B G E H I F C A
G
FED
B C
A
IH
Note
The preorder traverse is also called Polish
notation (after the Polish logician Jan
Lukasiewicz) and the postorder traverse is
therefore called a backwards Polish
notation

27. Binary Trees
 Mathematical formulas can be represented by Binary Trees:
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preorder - * * + 4 2 7 + 9 3 / 8 – 4
inorder (4 + 2) * 7 * (9 + 3) – (8/(-4)) (adding parentheses)
postorder 4 2 + 7 * 9 3 + * 8 4 - / -
-8*
* /
-
4+ 7
4 2
+
9 3

28. Sorted Binary Tree
A tree T is a Sorted Binary Tree where
 v: T  V function to receive the value of a node
 T a Binary Tree
 T1 the left Sub Tree (also a Sorted Binary Trees)
 T2 the right Sub Tree (also a Sorted Binary Trees) such that:
v(root(T)) > v(t) for all nodes t in T1
v(root(T)) ≤ v(t) for all nodes t in T2
 If you traverse a Sorted Binary Tree inorder, than all values
are reached in a sorted way
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29. Sorted Binary Tree
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961
3 8
5
Example
Inorder: 1 3 5 6 7 8 9

30. Binary Tree
Definitions
The height of a Tree is the length of the path from the root to
the deepest node in the Tree:
h(T) = 0 if the Tree T is empty
h(T) = max(h(T1), h(T2)) + 1
where T1 is the left and T2 the right Sub Tree
The cardinality of a Tree T is the number n of elements in
the Tree that means:
n = card(T)
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31. Binary Tree
Example
The height of this tree is 4
The cardinality of this tree is 7
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7
961
3 8
5

32. Binary Tree
A Tree is called height-balancing if the Tree T is empty or if
| h(T1) – h (T2) | ≤ 1
where h is the height of a Tree and also the Sub Trees T1 and
T2 are height-balancing Trees
A Tree is called balanced if the Tree T is empty or if
| card(T1) – card(T2) | ≤ 1
and also the Sub Trees T1 and T2 are balanced
If a Tree T is balance he is also balanced in its height
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33. Sorted Binary Tree
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height-balancing Tree but not a balanced
height left tree = 2, height right tree = 3
 |h(T1) – h(T2) = |2 – 3| = 1
card left tree = 2, card right tree = 4
 |card(T1) – card(T2) = |2 – 4| = 2
Balanced Tree ( height-balancing)
height left tree = 2, height right tree = 2
 |h(T1) – h(T2) = |2 – 2| = 0
card left tree = 3, card right tree = 3
 |card(T1) – card(T2) = |3 – 3| = 0
7
961
3 8
5
751
2 6
4
3

34. Binary Search Tree
k a value and the
s search element
v(s) = k
T (Binary Tree) with t = root(T)
If T is empty than there is nothing to do
If v(t) = k than the search element is found
If not than
 Replace T with the left subtree T1 if k < v(t)  T = T1
 Replace T with the right subtree T2 if k ≥ v(t)  T = T2
The effort for the search is O(h(T))
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35. Binary Search Tree
Example
Search element k
k = 1
t = 5  1 < 5
t = 3  1 < 3
t = 1  1 = 1 element found!
k = 7
t = 5  7 > 5
t = 8  7 < 8
t = 6  7 > 6
t = 7  7 = 7 element found!
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7
961
3 8
5

36. Insert a element in a Sorted
Binary Tree
u a values with k = v(u) to be insert
T a tree with t the root of Tree T
 Create a new binary sorted tree U containing only the root
u
 If T is empty replace T by U
 Otherwise
 replace T with the left sub tree T1 if k < v(t)  T = T1
 replace T with the right sub tree T2 if k ≥ v(t)  T = T2
 The effort is also O(h(T))
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37. Insert a element in a Sorted
Binary Tree
Example
Insert in an empty Tree the elements 5, 1, 3, 2, 8, 4, 6, 7 and 9:
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5
1
3
2 4
9
8
7
6

38. Delete a node in a Sorted Binary
Tree
Be u ϵ T the node to be removed from Tree T
Be U the sub tree from T with root(U) = u
 If the left Sub Tree U1 of U is empty than replace U by U1
 If the right Sub Tree U2 of U is empty than replace U by U2
 If both Sub Trees are not empty than:
 Chose an u2 ϵ U2 such that v(u2) ≤ v(t) for all t ϵ U2
 u2 has to be removed from U2
 u2 replace the node u
The effort to delete a node in a Tree is O(h(T)) due to the
search for the minimal element u2
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39. Delete a node in a Sorted Binary
Tree
Example
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40. Other Tree Types
 2-3 tree
 2-3-4 tree
 AA tree
 AVL tree
 B-tree
 Elastic binary tree
 Random binary tree
 Red-black tree
 Self-balancing binary search tree
 Unrooted binary tree
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41. Any
questions?
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