CHAPTER 06 Compiled by: Dr. Mohammad Omar Alhawarat Trees

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<p>CS2320: Data structures &amp; Algorithms</p> <p>Chapter 06Compiled by: Dr. Mohammad Omar AlhawaratTrees1Definition of TreeA tree is a set of linked nodes, such that there is one and only one path from a unique node (called the root node) to every other node in the tree</p> <p>2Hierarchical OrganizationExample: File Directories</p> <p>33Tree TerminologyNodes at a given level are children of nodes of previous levelNode with children is the parent node of those childrenNodes with same parent are siblingsNode with no children is a leaf nodeThe only node with no parent is the root nodeAll others have one parent each44Example of a Tree A C B D E F Groot5Example of a Tree A C B D E F GrootIn a tree, every pair of linked nodes have a parent-child relationship (the parent is closer to the root)6Example of a Tree (Cont.) A C B D E F GrootFor example, C is a parent of G7Example of a Tree (Cont.) A C B D E F GrootE and F are children of D8Example of a Tree (Cont.) A C B D E F GrootThe root node is the only node that has no parent.9Example of a Tree (Cont.) A C B D E F GrootLeaf nodes (or leaves for short) have no children.10Subtrees A B C I K D E F J G HsubtreerootA subtree is a part of a tree that is a tree in itself11Subtrees (Cont.) A B C I K D E F J G HsubtreerootIt normally includes each node reachable from its root.12Subtrees (Cont.) A B C I K D E F J G HrootEven though this looks like a subtree in itself13Binary TreesA binary tree is a tree in which each node can only have up to two children14Not a Binary Tree A B C I K D E F J G Hroot15Example of a Binary Tree A B C I K E J G HrootThe links in a tree are often called edges16Levels A B C I K E J G Hrootlevel 0level 1level 2level 3The level of a node is the number of edges in the path from the root node to this node17Full Binary Tree B D Hroot I A E J K C F L M G N OIn a full binary tree, each node has two children except for the nodes on the last level, which are leaf nodes 18Complete Binary TreesA complete binary tree is a binary tree that is eithera full binary treeORa tree that would be a full binary tree but it is missing the rightmost nodes on the last level19Complete Binary Trees (Cont.) B D Hroot I A E C F G20Complete Binary Trees (Cont.) B D Hroot I A E J K C F L G21Full Binary Trees B D Hroot I A E J K C F L M G N OA full binary tree is also a complete binary tree.22The formula for the maximum number of nodes is derived from the fact that each node can have only two descendants. Given a height of the binary tree, H, the maximum number of nodes in the full binary tree is given as follows:</p> <p>Full Binary Trees (Cont.)23To Assure that a binary tree is Balanced one the following algorithms is used:</p> <p>AVL Trees.Red-Black Trees.2-3 Trees and the general case (B-Trees)B-Trees are used with Fetching data from Large Databases.Balanced Binary Trees24Binary TreesA binary tree is either empty or has the following form</p> <p>Where Tleft and Tright are binary trees</p> <p>2525Binary TreesEvery nonleaf in a full binary tree has exactly two childrenA complete binary tree is full to its next-to-last levelLeaves on last level filled from left to rightThe height of a binary tree with n nodes that is either complete or full is log2(n + 1)</p> <p>2626</p> <p>Binary Trees2727A tree is a set of nodes that either:is empty orhas a designated node, called the root, from which hierarchically descend zero or more subtrees, which are also trees.Recursive definition of a tree28Recursive definition of a tree (Cont.)</p> <p>29Tree TraversalFour meaningful orders in which to traverse a binary tree.PreorderInorderPostorderLevel order3030Tree Traversal (Cont.)</p> <p>3131In Preorder, the rootis visited before (pre)the subtrees traversalsIn Inorder, the root isvisited in-between left and right subtree traversalIn Postorder, the rootis visited after (post)the subtrees traversalsPreorder Traversal:Visit the rootTraverse left subtreeTraverse right subtreeInorder Traversal:Traverse left subtreeVisit the rootTraverse right subtreePostorder Traversal:Traverse left subtreeTraverse right subtreeVisit the rootTree Traversal (Cont.)32Tree Traversal (Cont.)Visiting a nodeProcessing the data within a nodeThis is the action performed on each node during traversal of a treeA traversal can pass through a node without visiting it at that momentFor a binary treeVisit the rootVisit all nodes in the root's left subtreeVisit all nodes in the root's right subtree3333Tree Traversal (Cont.)Preorder traversal: visit root before the subtrees</p> <p>3434Tree Traversal (Cont.)Inorder traversal: visit root between visiting the subtrees</p> <p>3535Tree Traversal (Cont.)Postorder traversal: visit root after visiting the subtrees</p> <p>These are examples of a depth-first traversal.3636Tree Traversal (Cont.)Level-order traversal: begin at the root, visit nodes one level at a time</p> <p>This is an example of a breadth-first traversal.3737Tree Traversals Example 1</p> <p>38Assume: visiting a node is printing its labelPreorder: 1 3 5 4 6 7 8 9 10 11 12Inorder:4 5 6 3 1 8 7 9 11 10 12Postorder:4 6 5 3 8 11 12 10 9 7 113119846571210Tree Traversals Example 239Assume: visiting a node is printing its dataPreorder: 15 8 2 6 3 711 10 12 14 20 27 22 30Inorder: 2 3 6 7 8 10 1112 14 15 20 22 27 30Postorder: 3 7 6 2 10 1412 11 8 22 30 27 20 1561582371110141220272230Tree Traversals Example 340Tree Traversals Example 4</p> <p>41Examples of Binary TreesExpression Trees</p> <p>4242Example: Expression TreesStore expressions in a binary treeLeaves of tree are operands (e.g., constants, variables)Other internal nodes are unary or binary operatorsUsed by compilers to parse and evaluate expressionsArithmetic, logic, etc.E.g., (a + b * c)+((d * e + f) * g)434343</p> <p>43Cpt S 223Washington State UniversityTree Traversal - SummaryLevel order traversal is sometimes called breadth-first.The other traversals are called depth-first.Traversal takes (n) in both breadth-first and depth-first.Memory usage in a perfect tree is (log n) in depth-first and (n) in breadth-first traversal.4444Questions?Questions ?4545</p>