1. Tree Data Structures

2. Topics to be discussed….
Trees Data Structures
Trees
Binary Search Trees
Tree traversal
Types of Binary Trees
Threaded binary trees
Applications

3. Trees Data Structures Tree Binary Tree back Tree Nodes
Each node can have 0 or more children
A node can have at most one parent
Binary tree
Tree with 0–2 children per node
Tree
Binary Tree
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4. Trees Root node Height Interior nodes Leaf nodes Terminology
Root  no parent
Leaf  no child
Interior  non-leaf
Height  distance from root to leaf
Root node
Height
Interior nodes
Leaf nodes

5. Level and Depth
Level 1 2 3 4
node (13)
degree of a node
leaf (terminal)
Non terminal
parent
children
sibling
degree of a tree (3)
ancestor
level of a node
height of a tree (4) 3 1 2 2 1 2 3 2 2 3 3 3 1 3 3 3 4 4 4

6. Arithmetic Expression Using BT+
inorder traversal
A / B * C * D + E
infix expression
preorder traversal
+ * * / A B C D E
prefix expression
postorder traversal
A B / C * D * E +
postfix expression
level order traversal
+ * E * D / C A B * E * D / C A B
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7. Binary Search Trees X Y Z back Key property Value at node
Smaller values in left subtree
Larger values in right subtree
Example
X > Y
X < Z X Y Z
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8. Two types of tree traversal are; Recursive Non recursive
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9. Recursive traversal back Preorder: node, left, right
Inorder: left, node, right
Postorder: left, right, node
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10. Non recursive traversal
In this stack is used to implement the non recursive traversal.
All nodes first put into the stack and then each node is processed according to traversal.
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11. Not a binary search tree
Binary Search Trees 5 10 10 2 45 5 30 5 45 30 2 25 45 2 25 30 10 25
Not a binary search tree
Binary search trees

12. Example Binary Searches
Find (root, 25 ) 10 5
10 < 25, right
30 > 25, left
25 = 25, found
5 < 25, right
45 > 25, left
30 > 25, left
10 < 25, right
25 = 25, found 5 30 2 45 2 25 45 30 10 25

13. Degenerate binary tree
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
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14. Threaded Binary Trees back
Two many null pointers in current representation of binary trees n: number of nodes number of non-null links: n total links: 2n null links: 2n-(n-1)=n+1
Replace these null pointers with some useful “threads”.
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15. Applications
Pre-order traversal while duplicating nodes and values can make a complete duplicate of a binary tree. It can also be used to make a prefix expression (Polish notation) from expression trees: traverse the expression tree pre-orderly.
In-order traversal is very commonly used on binary search trees because it returns values from the underlying set in order, according to the comparator that set up the binary search tree (hence the name).
Post-order traversal while deleting or freeing nodes and values can delete or free an entire binary tree. It can also generate a postfix representation of a binary tree.
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16. Thank You