Algorithms and Data Structures

N.Wirth. Algorithms and Data Structures. Oberon version 198
occupies k words and that each pointer occupies one word of storage. What is the gain in storage when
using a binary tree versus an n-ary tree?
4.15. Assume that a tree is built upon the following definition of a recursive data structure (see Exercise
4.1). Formulate a procedure to find an element with a given key x and to perform an operation P on this
element.
RECTYPE Tree = RECORD x: INTEGER;
left, right: Tree
END
4.16. In a certain file system a directory of all files is organized as an ordered binary tree. Each node
denotes a file and specifies the file name and, among other things the date of its last access, encoded as an
integer. Write a program that traverses the tree and deletes all files whose last access was before a certain
date.
4.17. In a tree structure the frequency of access of each element is measured empirically by attributing to
each node an access count. At certain intervals of time, the tree organization is updated by traversing the
tree and generating a new tree by using the program of straight search and insertion, and inserting the keys
in the order of decreasing frequency count. Write a program that performs this reorganization. Is the
average path length of this tree equal to, worse, or much worse than that of an optimal tree?
4.18. The method of analyzing the tree insertion algorithm described in Sect. 4.5 can also be used to
compute the expected numbers C n of comparisons and M n of moves (exchanges) which are performed by
Quicksort, sorting n elements of an array, assuming that all n! permutations of the n keys 1, 2, ... n are
equally likely. Find the analogy and determine C n and M n .
4.19. Draw the balanced tree with 12 nodes which has the maximum height of all 12-node balanced trees.
In which sequence do the nodes have to be inserted so that the AVL-insertion procedure generates this
tree?
4.20. Find a sequence of n insertion keys so that the procedure for insertion in an AVL-tree performs
each of the four rebalancing acts (LL, LR, RR, RL) at least once. What is the minimal length n for such a
sequence?
4.21. Find a balanced tree with keys 1 ... n and a permutation of these keys so that, when applied to the
deletion procedure for AVL-trees, it performs each of the four rebalancing routines at least once. What is
the sequence with minimal length n?
4.22. What is the average path length of the Fibonacci-tree T n ?
4.23. Write a program that generates a nearly optimal tree according to the algorithm based on the
selection of a centroid as root.
4.24. Assume that the keys 1, 2, 3, ... are inserted into an empty B-tree of order 2. Which keys cause
page splits to occur? Which keys cause the height of the tree to increase? If the keys are deleted in the
same order, which keys cause pages to be merged (and disposed) and which keys cause the height to
decrease? Answer the question for (a) a deletion scheme using balancing, and (b) a scheme whithout
balancing (upon underflow, a single item only is fetched from a neighboring page).
4.25. Write a program for the search, insertion, and deletion of keys in a binary B-tree. Use the node
type and the insertion scheme shown above for the binary B-tree.
4.26. Find a sequence of insertion keys which, starting from the empty symmetric binary B-tree, causes