# Trees

Summary: In this laboratory, you will further explore issues of recursion over trees introduced in the reading on pairs and pair structures and continued in the reading on trees.

## Preparation

a. Make sure that you have the reading on pairs and pair structures and the reading on trees open in separate tabs and windows.

b. Make sure that you have a piece of paper and writing instrument handy.

## Exercises

### Exercise 1: A Color Tree Predicate

As you may recall from the reading, we have defined color trees as follows.

• Any color is a color tree.
• If `t1` and `t2` are color trees, then so is ```(cons t1 t2)```.
• Nothing else is a color tree.

Using this recursive definition, write a procedure, `(color-tree? val)` that returns true (`#t`) if val is a color tree and false (`#f`) otherwise.

### Exercise 2: Rendering Color Trees

Recall the `render-color-tree!` procedure from the reading.

```;;; Procedure:
;;;   render-color-tree!
;;; Parameters:
;;;   ctree, a tree of colors
;;;   image, an image
;;;   left, an integer
;;;   top, an integer
;;;   width, an integer
;;;   height, an integer
;;; Purpose:
;;;   Render the tree into the portion of the image bounded at
;;;   the left by left, at the top by top, and with the specified
;;;   width and height.
;;; Produces:
;;;   [Nothing; Called for the side effect.]
;;; Preconditions:
;;;   [The usual]
;;; Postconditions:
;;;   The tree has now been rendered.
(define render-color-tree!
(lambda (ctree image left top width height)
(cond
((pair? ctree)
(render-color-tree! (car ctree) image
left top
(/ width 2) height)
(render-color-tree! (cdr ctree) image
(+ left (/ width 2)) top
(/ width 2) height))
(else
(image-select-rectangle! image REPLACE
left top width height)
(context-set-fgcolor! ctree)
(image-fill-selection! image)
(context-update-displays!)
(image-select-nothing! image)))))
```

a. Add `render-color-tree!` to your definitions window.

b. Create a new 200x200 image named `canvas`.

c. What effect do you expect the following instruction to have?

````>` `(render-color-tree! "blue" canvas 0 0 200 200)`
```

e. What effect do you expect the following instruction to have?

````>` `(render-color-tree! (cons "black" "red") canvas 0 0 200 200)`
```

g. What effect do you expect the following instruction to have?

````>` `(render-color-tree! (cons (cons "green" "yellow") "orange") canvas 0 0 200 200)`
```

i. What effect do you expect the following instruction to have?

````>` `(render-color-tree! (cons "red" (cons (cons "blue" "purple") "black")) canvas 0 0 200 200)`
```

k. Create a color tree that you might be able to render in some interesting fashion.

### Exercise 3: Changing Orientation

Rewrite `render-color-tree!` so that it splits the image vertically rather than horizontally.

### Exercise 4: Preconditions

a. What preconditions should `render-color-tree!` have?

b. Use the `color-tree?` predicate from an earlier exercise to rewrite `render-color-tree!` so that it reports an appropriate error if its preconditions are not met. Use husk-and-kernel style, checking the preconditions in the husk.

c. Some programmers consider it inappropriate to scan a tree twice, once to make sure that it's valid and once to use the tree. Rewrite `render-color-tree!` so that it checks for and reports errors only when it is at one of the non-pair values.

### Exercise 5: Counting Cons Cells

Let's explore trees more generally. As you've noted, trees are built from cons cells (pairs). Sometimes it is useful to find out how many cons cells are in a tree.

a. Define and test a procedure named `cons-cell-count` that takes any Scheme value and determines how many boxes would appear in its box-and-pointer diagram. (The data structure that is represented by such a box, or the region of a computer's memory in which such a structure is stored is called a cons cell. Every time the `cons` procedure is used, explicitly or implicitly, in the construction of a Scheme value, a new cons cell is allocated, to store information about the car and the cdr. Thus `cons-cell-count` also tallies the number of times `cons` was invoked during the construction of its argument.)

For example, the structure in the following box-and-pointer diagram contains seven cons-cells, so when you apply `cons-cell-count` to that structure, it should return 7. On the other hand, the string `"sample"` contains no cons-cells, so the value of `(cons-cell-count "sample")` is 0.

In answering this question, you should consider whether each value, in turn, is a pair using the `pair?` predicate.

b. Use `cons-cell-count` to find out how many cons cells are needed to construct the list

``` (0 (1 (2 (3 (4))))) ```

See the notes at the end of the lab if you have trouble creating that list.

c. Draw a box-and-pointer diagram of this list to check the answer.

## Explorations

### Exploration 1: Building Randomized Color Trees

One potentially interesting way to experiment with color trees is to have a procedure build those trees. How? We might provide the procedure with an intended size of the tree.

• If the number of colors is 1, we return a randomly-selected color.
• If the number of colors is 2, we cons together two trees of size 1.
• Otherwise, we pick a random size for the left subtree (the size should be at least 1 and strictly less than the intended size of the whole tree). We then recursively build a tree of that size and another tree of an appropriate size and then cons them together.

a. Using this technique, write a procedure, `(random-bw-tree size)`, that randomly builds a black and white tree of the appropriate size.

b. Using this technique, write a procedure, `(random-color-tree size colors)`, that builds a random color tree of the desired size, randomly selecting the color from the list `colors`.

### Exploration 2: Color Trees, Revisited: Splitting the Space Horizontally and Vertically

Create a new version of `render-color-tree!` that has an extra parameter, `vsplit?`, a Boolean value. If the Boolean is true, when `render-color-tree!` should split the area vertically (as it currently does). If the Boolean is false, `render-color-tree!` should split the area horizontally. In both cases, the recursive calls should negate that Boolean value so that the splits alternate between vertical and horizontal.

## For Those with Extra Time

If you find that you have extra time, you might want to attempt one or more of the following problems.

### Extra 1: Cons Cells vs. Values

As you may recall, a tree is either (a) a non-pair value or (b) the cons of two trees. In the reading, you saw a procedure that counted the number of values in a tree. In this lab, you wrote a procedure that counted the number of cons cells (pairs) in a tree. What is the relationship between the numbers returned by those two procedures?

### Extra 2: Searching for Values

a. Write a procedure, ```(color-tree-contains? ctree color)```, that determines whether `color` appears anywhere in `ctree`. (We may have written a similar procedure during the class discussion. If so, you can start with that procedure.)

b. Rewrite the procedure to determine if a color nearby to `color` appears somewhere in `ctree`. (You might say that two colors are nearby if their red, green, and blue components all differ by less than sixteen.)

c. Write `(tree-contains tree value)`, a generalized version of `color-tree-contains` that works with a heterogeneous tree (that is, a tree that contains multiple kinds of values).

## Notes

### Notes on Exercise 5

If, for some reason, you are having trouble creating the list

`(0 (1 (2 (3 (4)))))`

try

`(list 0 (list 1 (list 2 (list 3 (list 4)))))`

Samuel A. Rebelsky, rebelsky@grinnell.edu

Copyright (c) 2007-9 Janet Davis, Matthew Kluber, Samuel A. Rebelsky, and Jerod Weinman. (Selected materials copyright by John David Stone and Henry Walker and used by permission.)

This material is based upon work partially supported by the National Science Foundation under Grant No. CCLI-0633090. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

This work is licensed under a Creative Commons Attribution-NonCommercial 2.5 License. To view a copy of this license, visit `http://creativecommons.org/licenses/by-nc/2.5/` or send a letter to Creative Commons, 543 Howard Street, 5th Floor, San Francisco, California, 94105, USA.