c36204 - 2 Trees 2 Trees 2.1 Preliminaries 2.2 Binary trees...

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Unformatted text preview: 2. Trees 2. Trees 2.1 Preliminaries 2.2 Binary trees 2.3 Binary search trees 2.4 AVL and Splay trees 2.5 B-trees CS340 Fall 2010 1 2.1 Preliminaries 2.1 Preliminaries A B C D E F G H I J K L M N P Q Root Leaves Height=3 CS340 Fall 2010 2 2.1 Preliminaries Terms • Child • Parent • Sibling : nodes that share a common parent. • Leaf : a node with no children. • Node depth : the number of links on the path from the root to the node. • Tree height : the depth of the deepest node. CS340 Fall 2010 3 2.1 Preliminaries Recursive view A tree is either : • Empty • Contains a root and N subtrees (N ≥ 0). CS340 Fall 2010 4 2.1 Preliminaries Implementation of Trees • The tree is a collection of nodes : struct TreeNode { Object element; TreeNode * firstChild; TreeNode * nextSibling; } ; • The tree stores a reference to the root node , which is the starting point. CS340 Fall 2010 5 2.1 Preliminaries Implementation of Trees A B C D E F G H I J K L M N P Q Root Height=3 CS340 Fall 2010 6 2.1 Preliminaries Tree Traversals // List a directory in a hierarchical file system void FileSystem::listAll(int depth = 0) const { printName( depth ); if (isDirectory()) for each file c in this directory(for each child) c.listAll(depth + 1); } ; // Calculate the size of a directory void FileSystem::size( ) const { int totalSize = sizeOfThisFile(); if (isDirectory()) for each file c in this directory(for each child) totalSize += c.size(); return totalSize; } ; CS340 Fall 2010 7 2.2 Binary Trees 2.2 Binary Trees Recursive view A binary tree is either : • Empty • Contains a root and N binary subtrees (0 ≤ N ≤ 2). CS340 Fall 2010 8 2.2 Binary Trees Implementation struct BinaryNode { Object element; // the data in the node BinaryNode * left; // left child BinaryNode * right; // right child } ; CS340 Fall 2010 9 2.2 Binary Trees Expression trees • Inorder traversal ⇒ infix notation. • postorder traversal ⇒ postfix notation. • preorder traversal ⇒ prefix notation. + + a * b c * + * d e f g CS340 Fall 2010 10 2.2 Binary Trees Constructing an expression tree Converting a postfix expression into an expression tree : • Same principle as evaluating an expression in postfix notation. • Use a stack of pointers. For infix expression : Step 1 : Use the algorithm seen in chapter 3 to convert the expression in postfix notation. Step 2 : Use the algorithm above to produce the corresponding expression tree. CS340 Fall 2010 11 2.3 Binary Search Trees 2.3 Binary Search Trees Goals : • Binary search supports Find in O (log N ) worst-case time, but Insert and Remove are O ( N ) . • Would like to support all three operations in O (log N ) worst-case time. • Today’s result : can support all three operations in O (log N ) average-case time....
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c36204 - 2 Trees 2 Trees 2.1 Preliminaries 2.2 Binary trees...

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