CIT208:Data Structures Algorithms II By Dr. E.K. Olatunji Computer Science Programme College of Computing and Communication Studies Bowen University, Iwo, Osun State, Nigeria March 2020 1 Tree Data Structure • References • 1. btechsmartclass.com/ds • 2. www.cs.cmu.edu/ • 3. computer Science by CS French • 4. tutorialpoint.com Tree Data Structure • A tree data structure (TDS) is a collection of data (nodes) which is organized in a hierarchical structure. • In TDS, every element is called a node • A node in TDS stores the actual data of that particular element and links to next element in the hierachical structure • It is non linear data structure unlike arrays, linked list, stack • Searching for element in a TDS is much faster than doing the same in array or linked list • A sample TDS is as shown in the next slide Sample TDS A B C D E K F G L • Figure XX1: A sample TDS Application of TDS • Storing of info that naturally form an hierachy, e.g the file system on a computer. Folders in OS are organized using TDS • In compiler during compilation, every expression (arith, logical, etc) is converted into a syntax tree format e,g, syntax tree of (a+b) *c is as shown below (next slide) • It is also used in auto corrector and spell checker • Assignment • Describe clearly 5 more applications of TDS Application of TDS Contd • Syntax trees of Algebraic Expressions * + • cc aa + * b c • aa bb b c • Fig xx2:Syntax tree for (a+b)*c (ii) a+ b*c Tree Terminology • An Example of tree • • Figure xx3: Example of a Tree data structure for Illustration • Source:http://btechsmartclass.com/data_structures/tree-terminology.html Tree Terminology Contd • Root Node – This is the first node in a TDS. It is the origin of a TDS – There is only one root node in every tree – In the sample TDS above (Fig xx 3) A is the root node • Edge – The connecting link or line between any 2 nodes • Parent – Any node apart from the root that has other nodes below it is called a parent node. In the above TDS, B is the parent of D,E, F • Child – A node below a given node connected downward by its edge. E.g D, E, F are children nodes of B; I and J are children of node E • Sibling – Nodes with the same parents • Leaf – A node without any child node. Also called a terminal node or External node e.g D, J, H • Internal Node – A node that has at least one child. Also called internal or non-terminal node Tree Terminology Contd • Degree – This is the total no of children a node has. The highest degree of a node among all nodes in a tree is called the degree of the Tree – E.g; the degrees of A & C are 2 while the degree of B is 3; and degree of D is zero. The degree of the tree is 3 • Level: – Level of a node represents d generation of a node. The root node is said to be at level 0; children node of the root are at level 1 & the children of the node at level 1 will be at level 2, etc. – For e.g; A is at level 0, B & C at level 1 while I & J are at level 3 • Height – In any tree, the “height” of a tree is the total no of edges/links from leaf to that node in its longest path. In a tree, the height of the root node is said to be the height of the tree. – The height of any leaf node is zero. From the sample tree, height of A is 3, which is the height of the tree; heights of B & C are 2 while the height of K is zero. Tree Terminology Contd • Depth – In a TDS, the total no of edges from the root node to a particular node is called the depth of the node. In a tree, the total no of edges from root node to a leaf node in the longest path is said to be the depth of the tree. – The depth of the root node is zero.. In the diagram, the depth of the tree is 3, depth of I, J, and K are 3; but the depth of C is 2. • Path – In a TDS, the sequence of nodes and edges from one node to the other node is called the path between the 2 nodes. For e.g; the path between A & J in the diagram above is A-B-E-J, The path between B & J is B-E-J. – The length of a path is the total no of nodes in that path. In the example above, the path A-B- E-J has length 4 • Sub Tree – This is a node together with all its descendants, itself being a child of another node (possibly the root node). – for example, in the diagram, B and all its descendants form a sub-tree; likewise, E and C form sub-trees with their descendants • Traversing – This is passing thru nodes in a specified order Types of Tree DS • A General Tree – This is a tree in which each node may have zero or more children. – It is used to model application such as file systems – Its Diagram here • Binary tree (BT) – It is a specialized case of a general tree – In a BT, each node cannot have more than 2 child nodes – Its diagram here • Full Binary Tree – This is a BT in which each node has exactly zero or 2 children. – In a Full BT, there is no node with exactly one child – Its diagram here • Complete Binary Tree – A complete BT is one which is completely filled from left to right (with possible exception of the bottom level). – Its diagram here Binary Search Tree (BST) • Binary search tree (BST) – It is a binary tree – A left child node must have a data value smaller than its parent node and the right child node must have a value greater than it s parent node. For e.g: – ** its Diagram here • Bb Constructing a BST • We assume the following data are to be entered into a tree in this order: 27, 14, 35, 10, 31, 19, 42 • i) 27 is the first data item to be placed into the tree, its node is therefore the root node, depicted as below: 27 • ii) Next we add 14 to the tree, using the rule ‘lower number to the left of the root and higher number to the right of the root/parent’; since 14 < its parent, i.e 27; the tree now looks like the one below 27 • 14 Constructing a BST • We assume the following data are to be entered into a tree in this order: 27, 14, 35, 10, 31, 19, 42 • iii) The next item is 35, which is larger than the root, so we add it to the right of its parent, thus the tree becomes • 27 14 35 • iv) Constructing a BST • We assume the following data are to be entered into a tree in this order: 27, 14, 35, 10, 31, 19, 42 • iv) we continue like this and the final tree is as shown here: 27 • * 14 35 10 19 31 42 • Figure yy1: A BST Constructing a BST Contd • Note: • The left most node contains the smallest value, while the rightmost node contains the highest value. • An important property of a BST is that an in- order traversal on it will always visit the nodes of the tree in a sorted order Basic Operation on a BST • Insert • Search • Traversal: – Pre-order – traverses a tree in a pre-order manner – In-order – traverses a tree in an in-order manner – Post-order – traverses a tree in a post-order manner Traversals in BST • This is the process of visiting all the nodes of a tree in order to process them, e.g printing all the values • There are 3 standardard traversal orders, each relating to when a node’s value is processed to when the value of its sub-trees are processed • The 3 traversals are: – Pre-order: processes the value in the current node 1st (NLR), then its left and right sub trees – In-order: Processes the left sub-tree 1st, then the value of its current node, then its right sub-tree (LNR) – Post-order: processes the left and right sub-trees 1st, then the value in the current node (LRN) • Thus the traversal order is determined by when a node’s value is processed compared to when the values in its sub-trees are processed: pre(before), in (in between) and post(after) Traversals in BST A B C D E F G • Figure xx4: Sample tree illustrating traversal orders In-order Traversal • In-order: – Left sub tree 1st, Root Next, Finally Right sub Tree (LNR) A B C D E F G – For example, from Figure xx4?? The in-order traversal will produce: DBEAFCG – Used if we need to produce an increasingly ordered list of the value stored in a tree Pre-order Traversal • Pre-order • NLR-Parent/Root node 1st, then Left sub tree and finally Right sub tree A B C D E F G • For e.g, from figure xx4?? A pre-order traversal will produce: ABDECFG • Can be used to make a prefix (polish notation) from expression tree; that is traverse the tree pre-orderly Post order Traversal • Post order _ – LRN: Left sub tree 1st, then Right sub tree next and then Root/parent node last A B C D E F G – For e.g; from figure xx4?? A post-order traversal will produce: DEBFGCA Post order Traversal • Post order _ – Can be used to generate postfix representation of a binary tree – If a tree represent the structure of an arithmetic expression with nodes representing operators and sub-tree representing operands, then a post-order traversal is best adapted to the problem (operands must be evaluated b4 their associated operator) – Assignment1 • What are the outputs of traversing the BST constructed in Figure yy1 (slide 15?)? – Assignment2? Tree DS • End of Lecture? GRAPH DATA STRUCTURE • It is a non-linear DS • It contains a set of points known as nodes( or vertices ) and a set of links or lines known as edges (or Arcs) which connect the vertices • Thus Graph can be defined as a collection of vertices (/nodes) and arcs (/edges) which connect the vertex • Generally, a graph G is represented as G = (V, E), where V is a set of vertices and E a set of edges Graph DS Contd • Example • The following is a graph with 5 vertices and 7 edges A B E C D • Figure 1: • The points A.