tree basics to advanced all concepts part 1
- 1. CSC303: Data Structures and Algorithms Audience: B. Tech. (CSE), Semester III Instructor: Dr. Mamata Dalui The study materials/presentations used in this course are solely meant for academic purposes and collected from different available course materials, slides, books, etc. Assistant Professor Department of Computer Science and Engineering National Institute of Technology Durgapur
- 3. • Faster than linear lists • More natural fit for some kinds of data Why a tree?
- 4. Example Tree root Activities Teaching Research Faculty Home Page Projects Publications CSE2003 CSC303
- 5. Tree • Definition (recursive): A tree is a finite set of one or more nodes such that • There is a specially designated node called root. • The remaining nodes are partitioned into n>=0 disjoint set T1,…,Tn, where each of these sets is a tree. T1,…,Tn are called the subtrees of the root. • Every node in the tree is the root of some subtree r T2 T1 Tn Root (designated node) …
- 6. • The root of each subtree T1, T2 …,Tn is said to be a child of r and r is the parent. • Trees are drawn like graphs, where there is an edge (sometimes directed) from a node to each of its children. Tree CSC303 Dept of CSE, NIT Durgapur 2020-21(Odd) r T2 T1 Tn Root (designated node) … level 0 level 1 level 2 level 3 A B C D E F G H I J K L M Subtrees Root
- 7. Some Terminologies • Node: A node of the tree stores the item of information plus the links to other nodes. Example: A, B, C, …, M are the nodes of this tree. • Degree: The maximum number of subtrees of a node is called the degree of the node. Example: Degree of node A is 3, that of B is 2 and C is 1. Degrees of the nodes E, J, K, L, G, M & I all are 0. • Degree of a tree: The maximum of the degree of the nodes in the tree is called the degree of the tree. Example: Here, degree of the tree is 3. Tree level 0 level 1 level 2 level 3 A B C D E F G H I J K L M Subtrees Root
- 8. Some Terminologies (cont’d) • Terminal nodes (or leaf): The nodes of the tree that have degree zero, are called terminal or leaf nodes. Example: Here, E, J, K, L, G, M & I are the leaf nodes. • Nonterminal nodes: The nodes that don’t belong to terminal nodes. Example: A, B, C, D, F, H are the non-terminal nodes. • Children: The roots of the subtrees of a node are the children of that node. The children of a node X are the immediate successors of X. Example: Here, B, C, D are the children of A. Tree level 0 level 1 level 2 level 3 A B C D E F G H I J K L M Subtrees Root
- 9. Some Terminology (cont’d) • Parent: The immediate predecessor of a node is the parent of the node. Example: A is the parent of B, C & D. Similarly, F is the parent of J, K, L and C is the parent of G. • Siblings: The children of the same parent are said to be siblings of each other. Example: B, C, & D are the siblings of each other. • Ancestors of a node: All the nodes along the path from the root to that node are the ancestors of that node. Example: Here, F, B, A are the ancestors of L. • Descendants of a node: All the nodes along the path from the node to any leaf node are the descendants of that node. Example: Here, H, M, I are the descendants of D. Tree level 0 level 1 level 2 level 3 A B C D E F G H I J K L M Subtrees Root
- 10. Some Terminology (cont’d) • The level of a node: The level is the rank in the hierarchy defined by letting the root to be at level zero. If a node is at level l, then it children are at level l+1. Example: Root A is at level 0, level of nodes B, C & D is 1. • Height (or depth): The maximum number of nodes in a path from the root node to a leaf is called the height of the tree. Thus h=lmax+1, where h is the height of the tree and lmax is the maximum level of the tree. Example: Here, the height of this tree is 4. Tree level 0 level 1 level 2 level 3 A B C D E F G H I J K L M Subtrees Root
- 11. Representation of Trees • List Representation • We can represent a tree as a list in which each of the subtrees is also a list • The root comes first, then followed by a list of sub-trees ( A ( B ( E, F ( J, K, L ) ), C ( G ), D ( H ( M ), I ) ) A B C D E F G H I J K L M Root
- 12. • Left Child-Right Sibling Representation A B C D E F G H I J K L M A B C D E F G H I J K L M Representation of Trees • In this representation, instead of having each node storing pointers to all of its children, a node will store pointer to just one of its child, the left child. • Apart from this the node will also store a pointer to its immediate right sibling. Original tree Left child right sibling representation of the tree
- 13. • Binary trees are characterized by the fact that any node can have at most two branches. • Definition (recursive): • A binary tree is a finite set of nodes that is either empty or consists of a root and two disjoint binary trees called the left subtree and the right subtree. A B C D F H I E G J K Binary Tree Binary tree
- 14. Varients of binary trees: (a) strictly binary tree (b) complete binary tree (c) almost complete binary tree (d) skewed tree (a) Strictly binary tree – If every non-leaf node in a binary tree is having non-empty left and right subtrees, the tree is strictly binary tree. A B C D E F G Level 0 Level 1 Level 2 Level 3 Binary Tree
- 15. (b) Complete binary tree – A complete binary tree of height h is the strictly binary tree all whose leaves are at level h. A B C D F H I Level 0 Level 1 Level 2 Level 3 E G J K L M N O Binary Tree
- 16. (c) Almost complete binary tree – A binary tree of height h is an almost complete binary tree if: 1. Any node nh at level less than h-1 has two children 2. For any node nh in the tree with a right descendant at level h, nh must have a left child and every left descendant of nh is either a leaf at level h or has two children A B C D F H I Level 0 Level 1 Level 2 Level 3 E G J Binary Tree
- 17. Binary Tree A B C D E F G (a) A B C D F H I E G J A B C D F H I E G J K A B C D F H I E G (a) Strictly binary tree, but not an almost complete binary tree, since leaf nodes are at levels 1, 2, 3 (Cond. 1 doesn’t satisfy) (b) (b) Strictly binary tree, but not an almost complete binary tree, since Cond 1 is true but Cond 2 fails as node A has right descendant at level 3 (J) but also has a left descendant at level 2 (E) (c) (c) Strictly binary tree, satisfies Cond 1 and 2, hence is an almost complete binary tree (d) Almost complete binary tree, but not a strictly binary tree, since node E has a left child but not right one (d)
- 18. (d) Skewed binary tree – A skewed binary tree is a type of binary tree in which all the nodes have only either one child or no child. A B C D Binary Tree Two special types of Skewed Binary trees: Left Skewed Binary Tree: Left skewed binary trees are those in which all the nodes are having either a left child or no child at all. These are left side dominated trees wherein all the right children remain as null. A B C D Right Skewed Binary Tree: Right skewed binary trees are those in which all the nodes are having either a right child or no child at all. These are right side dominated trees wherein all the left children remain as null. (a) Left-skewed binary tree (b) Right-skewed binary tree
- 19. CSC303 Dept of CSE, NIT Durgapur 2020-21(Odd) Thank You Questions?