Download to read offline
![Implementation of Binary Tree
struct tree
{
struct tree* lchild;
char data[10];
struct tree* rchild;
};](https://image.slidesharecdn.com/binaryandbtree-250914142710-23b1b4ad/85/Binary-and-B-tree-in-data-structure-pptx-43-320.jpg)
![Implementation of Binary Tree
struct tree
{
struct tree* lchild;
char data[10];
struct tree* rchild;
};](https://image.slidesharecdn.com/binaryandbtree-250914142710-23b1b4ad/75/Binary-and-B-tree-in-data-structure-pptx-43-2048.jpg)
Binary and B+tree in data structure.pptx
- 1. DATA STRUCTURES –Binary and B+ Tree
- 2. Trees - Introduction •A tree is a nonlinear hierarchical data structure that consists of nodes connected by edges. • Different tree data structures allow quicker and easier access to the data as it is a non-linear data structure.
- 3. Properties of Trees •There is one and only one path between every pair of vertices in a tree. • A tree with n vertices has n-1 edges. • A graph is a tree if and if only if it is minimally connected. • Any connected graph with n vertices and n-1 edges is a tree.
- 4. Tree terminologies • Node- A node is an entity that contains a key or value and pointers to its child nodes. – The last nodes of each path are called leaf nodes or external nodes that do not contain a link/pointer to child nodes. – The node having at least a child node is called an internal node. • Edge -It is the link between any two nodes. • Root - It is the topmost node of a tree.
- 5.
- 6. Root • The firstnode from where the tree originates is called as a root node. • In any tree, there must be only one root node. • We can never have multiple root nodes in a tree data structure.
- 7. Edge • The connectinglink between any two nodes is called as an edge. • In a tree with n number of nodes, there are exactly (n-1) number of edges.
- 8. Parent • The nodewhich has a branch from it to any other node is called as a parent node. • In other words, the node which has one or more children is called as a parent node. • In a tree, a parent node can have any number of child nodes.
- 9. Child • The nodewhich is a descendant of some node is called as a child node. • All the nodes except root node are child nodes.
- 10. Siblings • Nodes whichbelong to the same parent are called as siblings. • In other words, nodes with the same parent are sibling nodes.
- 11. Degree • Degree ofa node is the total number of children of that node. • Degree of a tree is the highest degree of a node among all the nodes in the tree.
- 12. Internal node • Thenode which has at least one child is called as an internal node. • Internal nodes are also called as non-terminal nodes. • Every non-leaf node is an internal node.
- 13. Leaf node • Thenode which does not have any child is called as a leaf node. • Leaf nodes are also called as external nodes or terminal nodes.
- 14. Level • In atree, each step from top to bottom is called as level of a tree. • The level count starts with 0 and increments by 1 at each level or step.
- 15. Height • Total numberof edges that lies on the longest path from any leaf node to a particular node is called as height of that node. • Height of a tree is the height of root node. • Height of all leaf nodes = 0
- 16. Depth • Total numberof edges from root node to a particular node is called as depth of that node. • Depth of a tree is the total number of edges from root node to a leaf node in the longest path. • Depth of the root node = 0 • The terms “level” and “depth” are used interchangeably.
- 17. Subtree • In atree, each child from a node forms a subtree recursively. • Every child node forms a subtree on its parent node.
- 18. Forest • A forestis a set of disjoint trees.
- 19. Binary Tree • Binarytree is a special tree data structure in which each node can have at most 2 children. • Thus, in a binary tree, Each node has either 0 child or 1 child or 2 children.
- 20. Unlabeled Binary Tree •A binary tree is unlabeled if its nodes are not assigned any label.
- 21. Example • Consider wewant to draw all the binary trees possible • Number of binary trees possible with 3 unlabeled nodes • = 2 x 3C3 / (3 + 1) • = 6C3 / 4 • = 5
- 22. Labeled Binary Tree •A binary tree is labelled if all its nodes are assigned a label.
- 23. Example • Consider wewant to draw all the binary trees possible with 3 labeled nodes. • Number of binary trees possible with 3 labeled nodes • = { 2 x 3C3 / (3 + 1) } x 3! • = { 6C3 / 4 } x 6 • = 5 x 6 • = 30
- 24. Types of BinaryTrees
- 25. Rooted Binary Tree •A rooted binary tree is a binary tree that satisfies the following 2 properties: – It has a root node. – Each node has at most 2 children.
- 26. Full/Strictly Binary Tree •A binary tree in which every node has either 0 or 2 children is called as a Full binary tree. • Full binary tree is also called as Strictly binary tree.
- 27. Complete /Perfect BinaryTree • A complete binary tree is a binary tree that satisfies the following 2 properties: – Every internal node has exactly 2 children. – All the leaf nodes are at the same level.
- 28. Almost Complete BinaryTree • An almost complete binary tree is a binary tree that satisfies the following 2 properties- – All the levels are completely filled except possibly the last level. – The last level must be strictly filled from left to right.
- 29. Skewed Binary Tree •A skewed binary tree is a binary tree that satisfies the following 2 properties- • All the nodes except one node has one and only one child. • The remaining node has no child. OR • A skewed binary tree is a binary tree of n nodes such that its depth is (n-1).
- 30. Tree Traversal • Inorder to perform any operation on a tree, you need to reach to the specific node. The tree traversal algorithm helps in visiting a required node in the tree. • Tree Traversal refers to the process of visiting each node in a tree data structure exactly once.
- 31. Tree traversal techniques TreeTraversal Depth First Traversal Breadth First Traversal Preorder Traversal Inorder Traversal Postorder Traversal
- 32. Depth First Traversal •Following three traversal techniques fall under Depth First Traversal- 1. Preorder Traversal 2. Inorder Traversal 3. Postorder Traversal
- 33. Preorder Traversal • Algorithm- –Visit the root – Traverse the left sub tree i.e. call Preorder (left sub tree) – Traverse the right sub tree i.e. call Preorder (right sub tree)
- 35. Inorder Traversal • Algorithm- –Traverse the left sub tree i.e. call Inorder (left sub tree) – Visit the root – Traverse the right sub tree i.e. call Inorder (right sub tree)
- 37. Postorder Traversal • Algorithm- –Traverse the left sub tree i.e. call Postorder (left sub tree) – Traverse the right sub tree i.e. call Postorder (right sub tree) – Visit the root
- 39. Breadth First Search •Breadth First Traversal of a tree prints all the nodes of a tree level by level. • Breadth First Traversal is also called as Level Order Traversal.
- 40. From a singletraversal it is not possible to construct unique binary tree. However any of the two traversals are given then the corresponding tree can be drawn uniquely: • Inorder and preorder • Inorder and postorder • Inorder and level order Building Binary Tree
- 41. Building Binary Tree Example1: Construct a binary tree from a given preorder and inorder sequence: Preorder: A B D G C E H I F Inorder: D G B A H E I C F Solution:
- 42.
- 43. Implementation of BinaryTree struct tree { struct tree* lchild; char data[10]; struct tree* rchild; };
- 44. Implementation of BinaryTree node* getnode() { node *temp ; temp = (node*) malloc(sizeof(node)); printf("n Enter Data: "); scanf("%s",temp->data); temp->lchild = NULL; temp->rchild = NULL; return temp; }
- 45.
- 46. B+ Tree A B+tree is a self-balancing tree data structure commonly used in databases and file systems for efficient storage and retrieval of large amounts of data. It is an extension of the B-tree, optimized for disk-based storage systems. Components of B+ Tree • Leaf nodes store all the key values and pointers to the actual data. • Internal nodes store only the keys that guide searches. • All leaf nodes are linked together, supporting efficient sequential and range queries.
- 47. Structure of B+Tree
- 48. B+ Tree Features Balanced:Auto-adjusts when data is added or removed, keeping search time efficient. Multi-level: Has a root, internal nodes, and leaf nodes (which store the data). Ordered: Maintains sorted keys, making range queries easy. High Fan-out: Each node has many children, keeping the tree short and fast. Cache-friendly: Works well with memory caches for better speed. Disk-efficient: Ideal for disk storage due to fast data access.
- 49. Searching record inB+ Tree • Start at the root node • Navigate through internal nodes based on key comparisons • Reach the appropriate leaf node • If the key exists, return the data; otherwise, report "record not found"
- 50. Searching record inB+ Tree Example: In the image below, in order to search for 58, traverse the root → internal node → reach the correct leaf node that contains 58 (if it exists).
- 51. Insertion in B+Tree Insertion in B+ Trees is done in three steps: 1. Navigate to the correct leaf node 2. Insert the new key in sorted order 3. If overflow occurs: • Split the leaf node • Push the middle key to the parent • This process may propagate up to the root
- 52. Deletion in B+Tree 1. Finding and removing the key from the appropriate leaf node 2. Rebalancing the tree to maintain B+ Tree properties: • Borrowing keys from siblings or • Merging nodes if necessary
- 53. Advantages of B+Tree • Data stored in a B+ tree can be accessed both sequentially and directly. • It takes an equal number of disk accesses to fetch records. • B+ trees have redundant search keys, and storing search keys repeatedly is not possible. • A B+ tree with 'l' levels can store more entries in its internal nodes compared to a B-tree having the same 'l' levels.
- 54. Disadvantages of B+Tree Slower exact match: Data is always in leaf nodes, increasing search path. More disk access: Requires reaching the leaf even for a single record. Complex updates: Insertion and deletion are more complex. Extra space: Leaf node links need additional pointers.
- 55.