- Improving the time or space complexity of an algorithm.
- Dealing with problems for which a recursively defined algorithm cannot readily be found.

In any recurrence for a divide-and-conquer algorithm, first make
explicit the costs of every part of the *divide*,
*conquer* (recurse), and *combine* steps, and relate
these costs to the relevant parts of the algorithm. Clearly state any
assumptions you make. Then maximally simplify the resulting
expressions before continuing.

To derive tight asymptotic bounds Θ(...), use the Master Theorem (MT) where possible. If the MT is applicable, then show in detail which case you apply, why it is applicable, and how you apply it. You can assume that the regularity condition holds, if need be. If the MT is not applicable, then first explain why that is so and then use any other suitable theorem or method seen in the course, again giving the full details of your reasoning.

The file binTrees.sml has
the beginning of a realisation of a polymorphic abstract datatype
(ADT) `'a bTree` for binary trees, specified on slides 6.19 to 6.24. The file trees.sml has sample binary trees for your tests
and the examples below. Add the following functions to that ADT, but
declare it as a *non-abstract* datatype, just to ease the
testing:

- The function
`preorder`returns the preorder walk, as a list, of a binary tree.

**Example:**`preorder smallIntTree = [1,2,5,7,3,6,10,13]`

**Requirements:**Do*not*use any help functions.

Can you make your function tail-recursive? Why / Why not? - The function
`preorder'`has the same specification as`preorder`.

**Requirements:**Use exactly*one*(local) help function, to be called`preorderAcc`, which must be obtained by introducing an accumulator. This technique is also known as performing a*descending generalisation*on the specification of`preorder`. Do*not*use any standard or library functions for lists, not even`@`.

Can you make your help function tail-recursive? Why / Why not? - The hidden function
`preorders`takes a*list*of binary trees as an argument and returns the concatenation of the preorder walks of the trees in that list. This specification was obtained by performing a*tupling generalisation*on the specification of`preorder`. This technique aims at reducing the number of recursive calls in each clause.

**Example:**`preorders [smallIntTree,anotherSmallIntTree] = [1,2,5,7,3,6,10,13,2,1,7,5,13,3,6,10]`

**Requirements:**Do*not*use any help functions. Do*not*use any standard or library functions for lists, not even`@`. Each clause must have*at most one*recursive call.

Can you make your function tail-recursive? Why / Why not? - The function
`preorder''`has the same specification as`preorder`.

**Requirement:**Use`preorders`as a (local) help function. - Discuss your answers to questions 1 to 4. In particular, if no
tail recursion in the strict sense has been achieved, would things
change with an inorder walk or a postorder walk? Would a
combination of the two generalisation techniques help to achieve
tail recursion in the strict sense with one of the three kinds of
walks?
- Give a recurrence for the running time
*T*of your`preorder`function on a*balanced*binary tree. Derive a tight asymptotic bound for*T*. - Give a recurrence for the running time
*T'*of your`preorderAcc`function on a*balanced*binary tree. Derive a tight asymptotic bound for*T'*. - Give a recurrence for the running time
*T''*of your`preorders`function on a*balanced*binary tree. Derive a tight asymptotic bound for*T''*. - Discuss your answers to questions 6 to 8.

- SML functions for Questions 1 to 4 above, in a .sml file,
executable under Moscow ML version 2.01, and documented, in
English, according to the coding
convention of the course.
- Replies, in English, to Questions 5 to 9 above, in a .txt or .pdf or .html file.