Trees (data structure)
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This document discusses trees as a data structure. It defines trees as structures containing nodes where each node can have zero or more children and at most one parent. Binary trees are defined as trees where each node has at most two children. The document discusses tree terminology like root, leaf, and height. It also covers binary search trees and their properties for storing and searching data, as well as algorithms for inserting and deleting nodes. Finally, it briefly discusses other types of trees like balanced search trees and parse trees.
Trees (data structure)
- 1. Trees (Data Structure) Trupti agrawal 1
- 2. Trees Data Structures 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 Trupti agrawal 2
- 3. Trees Terminology Root no parent Leaf no child Interior non-leaf Height distance from root to leaf Root node Interior nodes Height Leaf nodes Trupti agrawal 3
- 4. Binary Search Trees Key property Value at node Smaller values in left subtree Larger values in right subtree Example X > Y X < Z Y X Z Trupti agrawal 4
- 5. Binary Search Trees Examples Binary search trees Not a binary search tree 5 10 30 2 25 45 5 10 45 2 25 30 5 10 30 2 25 45 Trupti agrawal 5
- 6. Binary Tree Implementation Class Node { int data; // Could be int, a class, etc Node *left, *right; // null if empty void insert ( int data ) { … } void delete ( int data ) { … } Node *find ( int data ) { … } … } Trupti agrawal 6
- 7. Iterative Search of Binary Tree Node *Find( Node *n, int key) { while (n != NULL) { if (n->data == key) // Found it return n; if (n->data > key) // In left subtree n = n->left; else // In right subtree n = n->right; } return null; } Node * n = Find( root, 5); Trupti agrawal 7
- 8. Recursive Search of Binary Tree Node *Find( Node *n, int key) { if (n == NULL) // Not found return( n ); else if (n->data == key) // Found it return( n ); else if (n->data > key) // In left subtree return Find( n->left, key ); else // In right subtree return Find( n->right, key ); } Node * n = Find( root, 5); Trupti agrawal 8
- 9. 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 Trupti agrawal 9
- 10. 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 Trupti agrawal 10
- 11. 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 Trupti agrawal 11
- 12. 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 Trupti agrawal 12
- 13. 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 Trupti agrawal 13
- 14. Binary Search Tree Construction How to build & maintain binary trees? Insertion Deletion Maintain key property (invariant) Smaller values in left subtree Larger values in right subtree Trupti agrawal 14
- 15. Binary Search Tree – Insertion Algorithm 1. Perform search for value X 2. Search will end at node Y (if X not in tree) 3. If X < Y, insert new leaf X as new left subtree for Y 4. If X > Y, insert new leaf X as new right subtree for Y Observations O( log(n) ) operation for balanced tree Insertions may unbalance tree Trupti agrawal 15
- 16. Example Insertion Insert ( 20 ) 5 10 30 2 25 45 10 < 20, right 30 > 20, left 25 > 20, left Insert 20 on left 20 Trupti agrawal 16
- 17. Binary Search Tree – Deletion Algorithm 1. Perform search for value X 2. If X is a leaf, delete X 3. Else // must delete internal node a) Replace with largest value Y on left subtree OR smallest value Z on right subtree b) Delete replacement value (Y or Z) from subtree Observation O( log(n) ) operation for balanced tree Deletions may unbalance tree Trupti agrawal 17
- 18. Example Deletion (Leaf) Delete ( 25 ) 5 10 30 2 25 45 10 < 25, right 30 > 25, left 25 = 25, delete 5 10 30 2 45 Trupti agrawal 18
- 19. 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 Trupti agrawal 19
- 20. 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 Trupti agrawal 20
- 21. 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) B-trees Generalization of 2/3 trees Each internal node has between k/2 and k children Each node has an array of pointers to children Widely used in databases Trupti agrawal 21
- 22. Other (Non-Search) Trees Parse trees Convert from textual representation to tree representation Textual program to tree Used extensively in compilers Tree representation of data E.g. HTML data can be represented as a tree called DOM (Document Object Model) tree XML Like HTML, but used to represent data Tree structured Trupti agrawal 22
- 23. Parse Trees Expressions, programs, etc can be represented by tree structures E.g. Arithmetic Expression Tree A-(C/5 * 2) + (D*5 % 4) + - % A * * 4 / 2 D 5 C 5 Trupti agrawal 23
- 24. Tree Traversal Goal: visit every node of a tree in-order traversal void Node::inOrder () { if (left != NULL) { cout << “(“; left->inOrder(); cout << “)”; } cout << data << endl; if (right != NULL) right->inOrder() }Output: A – C / 5 * 2 + D * 5 % 4 To disambiguate: print brackets + - % A * * 4 / 2 D 5 C 5 Trupti agrawal 24
- 25. Tree Traversal (contd.) pre-order and post-order: void Node::preOrder () { cout << data << endl; if (left != NULL) left->preOrder (); if (right != NULL) right->preOrder (); } void Node::postOrder () { if (left != NULL) left->preOrder (); if (right != NULL) right->preOrder (); cout << data << endl; } + - % A * * 4 / 2 D 5 C 5 Output: + - A * / C 5 2 % * D 5 4 Output: A C 5 / 2 * - D 5 * 4 % + Trupti agrawal 25
- 26. THANK YOU…!!! Trupti agrawal 26