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Tree Data Structures

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**Topics to be discussed….**

Trees Data Structures Trees Binary Search Trees Tree traversal Types of Binary Trees Threaded binary trees Applications

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**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 back

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**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

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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

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**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 back

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**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 back

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**Two types of tree traversal are; Recursive Non recursive**

back

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**Recursive traversal back Preorder: node, left, right**

Inorder: left, node, right Postorder: left, right, node back

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**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. back

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**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

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**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

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**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 back

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**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”. back

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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. back

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