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Why Tree is considered a non-linear data structure?
- 3. In computer science, a tree is a widely-used data structure that emulates a hierarchical tree structure with a set of linked nodes.
- 4. Nature View of a Tree branch e s leave s root
- 6. What is a Tree ⚫ A tree is a finite nonempty set of elements. ⚫ It is an abstract model of a hierarchical structure. ⚫ consists of nodes with a parent-child relation. ⚫ Applications: ⚫ Organization charts ⚫ File systems ⚫ Programming environments Computers”R”Us Sale s R& D Manufacturin g Laptop s Desktop s US International Europ e Asi a Canad a
- 8. Cont… Root-The top most Node. Children Parents Siblings- Have same parents. Leaf- Has no Child. RELATION OF TREE 1 3 2 4 6 5 8 7 11 10 9 8
- 9. 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 Tre Binary Tree 9
- 10. Tree s ⚫Terminology ⚫Root no parent ⚫Leaf no child ⚫Interior non-leaf ⚫Height distance from root to leaf Root node Leaf nodes Interior nodes Height 1
- 12. Tree Terminology ⚫Path: Traversal from node to node along the edges results in a sequence called path. ⚫Root: Node at the top of the tree. ⚫Parent: Any node, except root has exactly one edge running upward to another node. The node above it is called parent. ⚫Child: Any node may have one or more lines running downward to other nodes. Nodes below are children. ⚫Leaf: A node that has no children. ⚫Subtree: Any node can be considered to be the root of a subtree, which consists of its children
- 13. subtre e Tree Terminology ⚫ Root: node without parent (A) ⚫ Siblings: nodes share the same parent ⚫ Internal node: node with at least one child (A, B, C, F) ⚫ External node (leaf ): node without children (E, I, J, K, G, H, D) ⚫ Ancestors of a node: parent, grandparent, grand-grandparent, etc. ⚫ Descendant of a node: child, grandchild, grand-grandchild, etc. ⚫ Depth of a node: number of ancestors ⚫ Height of a tree: maximum depth of any node (3) ⚫ Degree of a node: the number of its children ⚫ Degree of a tree: the maximum number of its node. A B D C G H E F I J K Subtree: tree consisting of a node and its descendants
- 14. General Trees ⚫Trees store data in a hierarchical manner ⚫ Root node is A ⚫ A's children are B, C, and D ⚫ E, F, and D are leaves ⚫ Length of path from A to E is 2 edges D A C F B E Trees have layers of nodes, where some are higher, others are lower.
- 15. Logic of Tr Used to represent hierarchical data. BS(CS) 1ST tree e.g. BS(CS)1st Section (A) Section (B) CR 1 CR2 group 1 group 2 group 3 group 4 group 1 group 2 group 3 group 4 1
- 16. Logic of Tr Used to represent hierarchical data. BS(CS) 1ST tree e.g. BS(CS)1st Section (A) Section (B) CR 1 CR2 group 1 group 2 group 3 group 4 group 1 group 2 group 3 group 4 1
- 17. Cont… A Collection of entities called Nodes. Tree is a Non-Linear Data Structure. It’s a hierarchica Structure. TREE Nodes 1
- 18. Tree Traversal ⚫Unlike linear data structures (Array, Linked List, Queues, Stacks, etc) which have only one logical way to traverse them, trees can be traversed in different ways. ⚫Traversal is a process to visit all the nodes of a tree and may print their values too. Because, all nodes are connected via edges (links) we always start from the root (head) node.
- 19. Cont… ⚫ That is, we cannot randomly access a node in a tree. There are three ways which we use to traverse a tree ⚫ (a) Inorder (Left, Root, Right) : 4 2 5 1 3 (b) Preorder (Root, Left, Right) : 1 2 4 5 3 (c) Postorder (Left, Right, Root) : 4 5 2 3 1 ⚫
- 20. In-order Traversal ⚫In this traversal method, the left subtree is visited first, then the root and later the right sub- tree. We should always remember that every node may represent a subtree itself.
- 21. Inorder (Left, Root, Right)
- 22. Inorder (Left, Root, Right) ⚫We start from A, and following in-order traversal, we move to its left subtree B. B is also traversed in-order. The process goes on until all the nodes are visited. The output of inorder traversal of this tree will be − ⚫D → B → E → A → F → C → G
- 23. Inorder Algorithm ⚫Until all nodes are traversed − ⚫ Step 1 − Recursively traverse left subtree. ⚫ Step 2 − Visit root node. ⚫Step 3 − Recursively traverse right subtree.
- 24. Pre-order Traversal ⚫In this traversal method, the root node is visited first, then the left subtree and finally the right subtree.
- 25. Preorder (Root, Left, Right)
- 26. Preorder (Root, Left, Right) ⚫ We start from A, and following pre-order traversal, we first visit A itself and then move to its left subtree B. B is also traversed pre-order. The process goes on until all the nodes are visited. The output of pre- order traversal of this tree will be. ⚫ A → B → D → E → C → F → G
- 27. Algorithm ⚫Until all nodes are traversed − ⚫Step 1 − Visit root node. ⚫Step 2 − Recursively traverse left subtree. ⚫ Step 3 − Recursively traverse right subtree.
- 28. Post-order Traversal ⚫In this traversal method, the root node is visited last, hence the name. First we traverse the left subtree, then the right subtree and finally the root node.
- 29. Postorder (Left, Right, Root)
- 30. Postorder (Left, Right, Root) ⚫ We start from A, and following pre-order traversal, we first visit the left subtree B. B is also traversed post-order. The process goes on until all the nodes are visited. The output of post-order traversal of this tree will be. ⚫ D → E → B → F → G → C → A
- 31. Algorithm ⚫Until all nodes are traversed ⚫ Step 1 − Recursively traverse left subtree. ⚫ Step 2 − Recursively traverse right subtree. ⚫ Step 3 − Visit root node.
- 33. • If root is NULL • then create root node • return • If root exists then • compare the data with node.data • while until insertion position is located • If data is greater than node.data • goto right subtree • else • goto left subtree • endwhile • insert data
- 34. If root.data is equal to search.data return root else while data not found If data is greater than node.data goto right subtree else goto left subtree If data found return node endwhile return data not found end if
- 37. There are a large number of tree types, however three types specifically are used the majority of the time (both in real world development and in Computer Science courses). The types are: Binary trees B-Trees Heaps
- 38. Binary trees ⚫Binary tree follows the rule that each parent node has no more than 2 child nodes. This means that a binary tree can look like this: ⚫ OR
- 39. Strict Binary Tree ⚫Strict Binary Tree is a Binary tree in which root can have exactly two children or no children at all. That means, there can be 0 or 2 children of any node.
- 40. Complete Binary Tree ⚫Complete Binary Tree is a Strict Binary tree in which every leaf node is at same level. That means, there are equal number of children in right and left subtree for every node.
- 41. Extended Binary Tree ⚫An extended binary tree is a transformation of any binary tree into a complete binary tree. ⚫This transformation consists of replacing every null subtree of the original tree with “special nodes.” ⚫The nodes from the original tree are then called as internal nodes, while the “special nodes” are called as external nodes.
- 42. Extended Binary Tree A binary tree with special nodes replacing every null subtree. Every regular node has two children, and every special node has no children.
- 43. Binary Search Tree ⚫ The value at any node, • Greater than every value in left subtree. Smaller than every value in right subtree – Example ⚫Y > X ⚫Y < Z ⚫ Values in left sub tree less than parent ⚫ Values in right sub tree greater than parent ⚫ Fast searches in a Binary Search tree, maximum of log n comparisons.
- 44. Binary Search Trees ⚫Example s ⚫Binary search tree
- 45. B - Trees ⚫B – Tree is another type of search tree called B- Tree in which a node can store more than one value (key) and it can have more than two children. ⚫B-Tree can be defined as follows… ⚫ B-Tree is a self-balanced search tree with multiple keys in every node and more than two children for every node.
- 46. Operations on a B-Tree ⚫The following operations are performed on a B- Tree... ⚫Search ⚫Insertion ⚫Deletion
- 47. Heap Tree ⚫Heap is a special case of balanced binary tree data structure where the root-node key is compared with its children and arranged accordingly. If α has child node β then ⚫key(α) ≥ key(β) ⚫A heap can be of two types ⚫ Min-Heap ⚫ Max-Heap
- 48. Max-Heap ⚫Max-Heap − Where the value of the root node is greater than or equal to either of its children.
- 51. Max Heap Construction Algorithm ⚫Step 1 − Create a new node. ⚫ Step 2 − Assign new value to the node. ⚫Step 3 − Compare the value of this child node with its parent. ⚫Step 4 If − value of parent is less than child, then swap them. ⚫Step 5 − Repeat step 3 & 4 until Heap property holds.
- 52. Max Heap Deletion Algorithm ⚫Step 1 − Remove root node. ⚫Step 2 − Move the last element of last level to root. ⚫Step 3 − Compare the value of this child node with its parent. ⚫Step 4 If − value of parent is less than child, then swap them. ⚫Step 5 − Repeat step 3 & 4 until Heap property holds.
- 53. Min-Heap ⚫Min-Heap − Where the value of the root node is less than or equal to either of its children.