Figure 1835 Mystery program for Exercise 183

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186 A full node is a node with two children Prove that in a binary tree the number of full nodes plus 1 equals the number of leaves 187 How many NULL children are there in a binary tree of N nodes How many are in an M-ary tree of N nodes 188 Suppose that a binary tree has leaves l , , 4, , lM at depths d , , d 2 , , d M , respectively Prove that I = 1 2-d1 6 1 and determine when equality is true (known as Kraft's inequality)

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In Practice

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189 Write efficient functions (and give their Big-Oh running times) that take a pointer to a binary tree root T and compute a, the number of leaves in T b the number of nodes in T that contain one non-NULL child c the number of nodes in T that contain two non-NULLchildren 1810 Implement some of the recursive routines with tests that ensure that a recursive call is not made on a NULL subtree Compare the running time with identical routines that defer the test until the first line of the recursive function 1811 Rewrite the iterator class to throw an exception when f i r s t is applied to an empty tree Why might this be a bad idea Programming Projects 1812 A binary tree can be generated automatically for desktop publishing by a program You can write this program by assigning an x-y coordinate to each tree node, drawing a circle around each coordinate, and connecting each nonroot node to its parent Assume that you have a binary tree stored in memory and that each node has two extra data members for storing the coordinates Assume that (0, 0) is the top-left corner Do the following a The x-coordinate can be computed by assigning the inorder traversal number Write a routine to do so for each node in the tree b The y-coordinate can be computed by using the negative of the depth of the node Write a routine to do so for each node in the tree c In terms of some imaginary unit, what will be the dimensions of the picture Also determine how can you adjust the units so that the tree is always roughly two-thirds as high as it is wide

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d Prove that when this system is used, no lines cross and that for any node X, all elements in X's left subtree appear to the left of X , and all elements in X's right subtree appear to the right of X e Determine whether both coordinates can be computed in one recursive function f Write a general-purpose tree-drawing program to convert a tree into the following graph-assembler instructions (circles are numbered in the order in which they are drawn):

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circle i x, y ) ; drawline( i , j ) ;

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!i Draw circle with center i x , y)

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I / Connect circle i to circle j

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g Write a program that reads graph-assembler instructions and outputs the tree to your favorite device

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1813 If you are running on a Unix system, implement the du command

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19

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For large amounts of input, the linear access time of linked lists is prohibitive In this chapter we look at an alternative to the linked list: the binary search tree, a simple data structure that can be viewed as extending the binary search algorithm to allow insertions and deletions The running time for most operations is O(log N) on average Unfortunately, the worst-case time is O(N) per operation In this chapter, we show: the basic binary search tree, a method for adding order statistics (ie, the findKth operation), three different ways to eliminate the O(N) worst case (namely, the AVL tree, red-black tree, and AA-tree), implementation of the STL s e t and map, and use of the B-tree to search a large database quickly

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