1. Data Structures & Algorithm Design
Trees
Mais S. Al-Saoud
Computer Science Department
Karballa University
March 2015

2. Tree Definition Tree: a set of elements of the same type such that
it has a distinguished element called the root from which descend zero or more trees (subtrees)
Examples in real life:
Family tree
Table of contents of a book
Recursive! 2

3. Tree Terminology Nodes: the elements in the tree
Edges: connections between nodes
Root: the distinguished element that is the origin of the tree
There is only one root node in a tree
Leaf node: a node without an edge to another node
Interior node: a node that is not a leaf node
Empty tree has no nodes and no edges

4. Tree Terminology
Parent or predecessor: the node directly above in the hierarchy
A node can have only one parent
Child or successor: a node directly below in the hierarchy
Ancestors of a node: its parent, the parent of its parent, etc.
Descendants of a node: its children, the children of its children, etc.

5. Tree Terminology
Subtree of a node: consists of a child node and all its descendants
A subtree is itself a tree
A node may have many subtrees
If A is the root of a binary tree and B is the root of its left or right subtrees, then A is said to be the father of B and B is said to be the left/right son of A.
Two nodes are brothers if they are left and right sons of the same father.

6. Tree Terminology
Root
Subtrees of the root

7. Subtrees E
Subtrees of the node labeled E

8. Tree Terminology
Root
Interior nodes
Leaf nodes

9. Height of a Tree
A path is a sequence of edges leading from one node to another
Length of a path: number of edges on the path
Height of a (non-empty) tree : length of the longest path from the root to a leaf
(maximum level of any node)
What is the height of a tree that has only a root node?
By convention, the height of an empty tree is -1 9

10. Level and Degree of a Node
Level of a node : number of edges between root and node
Level of root node is 0
Level of a node that is not the root node is level of its parent + 1
Degree of a node: the number of children it has
Degree of a tree: the maximum of the degrees of the tree’s nodes

11. Level of a Node
Level 0
Level 1
Level 2
Level 3

12. Binary Trees
General tree: a tree each of whose nodes may have any number of children
n-ary tree: a tree each of whose nodes may have no more than n children
Binary tree: a tree each of whose nodes may have no more than 2 children
a binary tree is a tree with degree 2

13. Binary Trees Tree with 0–2 children per node
The children (if present) are called the left child and right child
a tree where every node has left and right subtrees is a binary tree

14. Binary Trees Strictly Binary Tree Complete Binary Tree
All non-leaf nodes have two branches
Complete Binary Tree
All its leaf nodes at the same level
Almost Complete Binary Tree
The last level is partially filled from left to right.

15. Binary Trees A B C D E F G H I J K
Not Complete BT, Strictly BT, Almost complete BT

16. Binary Tree Properties
The number of nodes n in a complete binary tree is at least n = 2h and at most n = 2(h + 1) − 1 where h is the height of the tree.
The number of leaf nodes L in a complete binary tree can be found using this formula: L = 2h where h is the height of the tree.
The number of nodes n in a perfect binary tree can also be found using this formula:
n = 2L − 1 where L is the number of leaf nodes in the tree.

17. Binary Tree Properties-Example

18. Binary Search Tree
Binary Search Tree is a binary tree in which nodes are arranged according to their values.
The left node has a value less than its parent
The right node has a values greater than its parent

19. Binary Search Tree Construction
How to build & maintain binary trees?
Insertion
Deletion
Maintain key property
Smaller values in left subtree
Larger values in right subtree

20. Declaration BT struct node { A node consists of three fields such as :
Left Child (LChild)
Information of the Node (Info)
Right Child (RChild)
struct node {
int info;
struct node *left;
struct node *right; };

21. Binary Search Tree – Insertion
Perform comparison for value X
Comparison will end at node Y (if X not in tree)
If X < Y, insert new leaf X as new left subtree for Y
If X > Y, insert new leaf X as new right subtree for Y

22. Not a binary search tree
Insertion- Examples 5 10 10 2 45 5 30 5 45 30 2 25 45 2 25 30 10 25
Binary search trees
Not a binary search tree

23. Insertion- Example Insert ( 20 ) 10 20 >10, right 20 < 30, left
Insert 20 as left leaf 5 30 2 25 45 20

24. Binary Search Tree Given the following sequence of numbers,
14, 15, 4, 9, 7, 18, 3, 5, 16, 4, 20, 17, 9, 14, 5
The following binary search tree can be constructed.

26. Algorithm for insert node in BST
Steps processing (by Recursion)
1.check if tree not contain any node
if Root = NULL
create node in tree
2.Else
if x< Root  Inf
Add the new value in the left subtree
Rootleft = insert(Rootleft, x);
3.Else
Add the new value in the right subtree
Rootright = insert(Rootright, x);

27. Insertion of node in BST
node * Insert(node *p, int x) { if(p==NULL) { p=make(x); return p; } else { if(x<p->inf) p->left=Insert(p->left, x); else p->right=Insert(p->right,x); return p; } }

28. Traversing :
Means visiting all the nodes of the tree, in some specific order for processing.
There are THREE ways of Traversing a Tree
1. Preoder Traversal
2. Inorder Traversal
3. Postorder Traversal
4. Levelorder Traversal

29. Preorder Traversal Method
Traversing a binary tree in preorder (Root – Left – Right) 1. Visit the root. 2. Traverse the left subtree in preorder. 3. Traverse the right subtree in preorder.

30. Ex 1
Ex 2

31. Preorder Traversal Method
void PreOrder(node *&h) { if (h!=NULL) cout<<h->inf; PreOrder(h->left); PreOrder(h->right); } }

32. Inorder Traversal Method
Traversing a binary tree in inorder
(Left – Root – Right)
1. Traverse the left subtree in inorder.
2. Visit the root.
3. Traverse the right subtree in inorder

33. Inorder Traversal Method
int InOrder(node *&h) { if (h!=NULL) InOrder(h->left); cout<<h->inf; InOrder(h->right); } }

34. Ex 1
Ex 2

35. Postorder Traversal Method
Traversing a binary tree in postorder
(Left – Right – Root)
1. Traverse the left subtree in postorder.
2. Traverse the right subtree in postorder.
3. Visit the root postorder

36. Ex 1
Ex 2

37. Postorder Traversal Method
int PostOrder(node *&h) { if (h!=NULL) PostOrder(h->left); PostOrder(h->right); cout<<h->inf; } }

38. Levelorder Traversal Method
levelorder Example (Visit = print)
Travers tree level by level

39. Traversals: Example