8.haftajava notlarıiçeriyordökumanlar.pdf
- 1. BMI2123 Data Structures Asst [email protected] Week 8:Trees So Far Arrays Now, familiar with algorithmic complexity analysis Order of running time Big-Oh function Stacks Queues A midterm exam 2 Today .. 3
- 2. Linear vs Nonlinear Data Structures Linear Data Structure: Data is organized in sequential (linear) order. Data elements are traversed one after the other. Only one element can be directly reached while traversing. Examples: Array, stack, linked lists, queue Nonlinear Data Structure: Data is not organized in sequential order. Data can be organized hierarchically or random order. Examples: Tree, graph, dictionaries, heaps What is a Tree? A tree is an abstract data type that stores elements hierarchically. A tree is a graph without cycles A nonlinear structure Recall: A data structure is said to be nonlinear if its elements form a hierarchical classification where, data items appear at various levels In software systems, a tree is an abstract model of a hierarchical structure such as lists, vectors, and sequences A tree consists of nodes with a parent-child relation 6 Recursive Nature of Trees A tree is hierarchical collection of nodes. One of the nodes, known as the root, has no parent. The links leaving a node (any number of links are allowed) point to child nodes. Trees are recursive structures. Each child node is itself the root of a subtree. At the bottom of the tree are leaf nodes, which have no children. root Recursive definition: A tree is either: empty (null), or a root node that contains: data, a left subtree, and a right subtree. (The left and/or right subtree could be empty.) Trees in Computer Science folders/files on a computer family genealogy; organizational charts
- 3. AI: decision trees compilers: parse tree a = (b + c) * d; cell phone T9 d + * a = c b Image source: https://images.datacamp.com/image/upload/v1677504957/decision_tree_for_heart_attack_prevention_2140 bd762d.png Trees in Computer Science(cont.) A parse tree Node: An object containing a data value and left/right children. Root: the node with no parents. There can be at most one root node in a tree Leaf: a node that has no children Branch: any internal node; neither the root nor a leaf parent: a node that refers to this one child: a node that this node refers to sibling: the nodes which are children of same parent A node p is an ancestor of node q if there exists a path from root to q and p appears on the path. The node q is called a descendant of p. Edge: the link from parent to child Subtree: the smaller tree of nodes on the left or right of the current node Tree Terminology Node: An object containing a data value and left/right children. Root: the node with no parents. There can be at most one root node in a tree A Leaf: a node that has no children E,J,K,H,I Branch: any internal node; neither the root nor a leaf B,C,D,F,G parent: a node that refers to this one B,C,D,F,G child: a node that this node refers to B,C,D,E,F,G,H,I,J,K sibling: the nodes which are children of same parent {B,C,D}, {E,F}, {H,I} Edge: the link from parent to child All links in the figure Tree Terminology Answers Tree Terminology(cont.) level or depth: length of the path from a root to a given node. The root node is at level zero. For example, (1 and 4) at the same level and (2 and 5) at the same level. The height of a tree is the number of links in the longest path from the root to a leaf. Empty tree Right-skewed tree One node
- 4. Binary Trees A tree is called binary tree if each node has zero child, one child or two children. Empty tree is also a valid binary tree. There are four major types of binary trees: Full Binary Tree - Every parent node has either two or no children. Perfect Binary Tree - Every parent node has exactly two child nodes and all the external nodes (leaf nodes) are at the same level. Complete Binary Tree - In this case, every level must be filled, and all the leaf elements lean towards the left. Balanced Binary Tree - A binary tree in which the height difference between the left and the right child nodes is either 0 or 1. Tree ADT It is useful to store collections of positions. For example, the children of a node in a tree can be presented to the user as such a list. We define position list, to be a list whose elements are tree positions. The real power of a tree position arises from its ability to access the neighboring elements of the tree. Given a position p of tree T, we define the following: 14 Properties of Binary Trees For the following properties, let us assume that the height of the tree is h. Also, assume that root node is at height zero. where n is the number of nodes in the tree, h is the height of the tree Representation of A Binary Tree A binary tree data structure is represented using two methods. Those methods are 1-Array Representation and 2-Linked List Representation Array Representation In array representation of binary tree, we use a one dimensional array (1-D Array) to represent a binary tree.
- 5. Array Representation(cont.) Waste of space problem! 2-Linked List Representation We use linked list to represent a binary tree. In a linked list, every node consists of three fields. First field, for storing left child address, second for storing actual data and third for storing right child address. Tree Traversal traversal: An examination of the elements of a tree. A pattern used in many tree algorithms and methods Common orderings for traversals: pre-order: process root node, then its left/right subtrees in-order: process left subtree, then root node, then right post-order: process left/right subtrees, then root node level-order: Access nodes level by level starting from root. (also known as Breadth First Search) 40 81 9 41 17 6 29 Root pre-order: ? in-order: ? post-order: ? level-order: ? Traversal trick To quickly generate a traversal: Trace a path around the tree. As you pass a node on the proper side, process it. pre-order: left side in-order: bottom post-order: right side pre-order: 17 41 29 6 9 81 40 in-order: 29 41 6 17 81 9 40 post-order: 29 6 41 81 40 9 17 40 81 9 41 17 6 29 overallRoot
- 6. Examples 3 86 9 15 42 27 48 12 39 5 Solution for the examples Arithmetic Expression Tree Binary tree associated with an arithmetic expression internal nodes: operators external nodes: operands Example: arithmetic expression tree for the expression (2 (a 1) (3 b)) 23 2 a 1 3 b Note: Properties of Proper Binary Trees Notation n number of nodes e number of external nodes i number of internal nodes h height 24 Properties: e = i + 1 n = 2e - 1 h i h (n - 1)/2 e 2h h log2 e h log2 (n + 1) - 1
- 7. Acknowledgement This material is prepared based on Data Structures and Algorithms in C++ 2e book and the lecture notes of Prof. Yung Yi from KAIST, South Korea. We thank to Michael T. Goodrich, Roberto Tamassia, David M. Mount who are the authors of the book and Prof. Yung Yi.