# 5  Binary Search Trees; Graphics and Interactions

## Binary Search Trees

We have the following data definition:

A Binary Search Tree is one of

• Empty Tree

• a Node

A Node consists of

• `data` of the type `String`,

• `left` Binary Search Tree,

• `right` Binary Search Tree.

Additionally, every `Node` has the property that all data in the `left` subtree contains strings that appear in the dictionary before the data in the `Node`, and all data in the `right` subtree are strings that appear in the dictionary after the `data` in this node. The string that is identical to the field `data` can be in either the `left` or in the `right` subtree -- it is not important.

The following are examples of such trees:

```             ppp                          mmm
------                      --------
/      \                    /        \
ggg        ttt                ddd          rrr
-----      -----              -----        -----
/     \    /     \            /     \      /     \
bbb      *  sss     *        aaa       kkk  *       xxx
-----       -----            -----     -----        -----
/     \     /     \          /     \   /     \      /     \
*       *   *       *        *       * *       *    *       *

Tree-1                            Tree-2
```

Your task is to design classes that represent this data as Java classes, and to design several methods to manipulate this data. You will also need an auxilliary classes that represent a list of `String`s. You may read more about binary search tree in HtDP.

## 5.1  Problem

Design the classes that represent binary search trees of `String`s. Make examples of data.

## 5.2  Problem

Design the method `same` that determines whether two binary search trees are the same, i.e. they have the same structure and contain the same `String`s.

## 5.3  Problem

Design the method `insert` that insert a `String` into a binary search tree, preserving the tree property stated earlier.

Java provides the following method for `String` comparison:

```// compare this String with that String lexicographically
// return <0  --- if 'this' is before 'that'
// return 0   --- if 'this' is the same String as 'that'
// return >0  --- if 'this' is after 'that'
int compareTo(String that) ...
```

If you do not understand how to do it, try some examples by hand, or read about the problem in HtDP.

## 5.4  Problem

Design the classes that represent a list of `String`s. Add the method `sort` that sorts the list in lexicographical order. Add the method `same` that determines whether two lists contain the same `String`s in the same order. You should just modify your solutions to the previous homework.

## 5.5  Problem

Design the method `inorder` that produces a list of `String`s that appear in the nodes of this tree, ordered so that all strings in the left subtree appear in the list before the data in the root node, and all strings in the right subtree appear in the list after the data in the root node.

For example, our two examples would produce the lists in the following order:

```Tree-1:  bbb ggg ppp sss ttt
Tree-2:  aaa ddd kkk mmm rrr xxx
```

The method consumes a list of `String`s, initially empty, that represents the list of strings that come after all the nodes in this subtree have been entered into the list. So, for our Tree-2, in the `Node` ddd, the given list of strings will contain (mmm rrr xxx). The nodes in this subtree, ddd, aaa, and kkk, still have to be added to the list. It is clear, that when we start at the node mmm, there is nothing in the list.

Again, make examples until you understand the problem. Follow the design recipe!

## 5.6  Problem

Design the method `contains` for both, the classes that represent the binary search trees, and the classes that represent lists of `Strings`. The method determines whether the given `String` appears in the tree or list repsectively.

## 5.7  Problem

Explore the power of the methods you designed through the following examples:

1. insert the same items into a binary search tree several times in different order, then produce the result from the `inorder` method and check that they are the same.

2. insert the same items into a list of `String`s, again several times in a different order, and sort these lists.

3. design a comparison between a binary search tree and a list of `String`s to determine whether they contain the same `String`s.

4. design the method `sameData` for the classes that represent binary
search trees, that determines whether two trees contain the same
`String`s, not necessarily organized the same way. For example, any pair of the following three trees would pass this test:

```      bb                  cc              aa
/    \              /    \          /    \
aa      cc          aa      *        *      bb
/  \    /  \        /  \                    /  \
*    *  *    *      *   bb                  *   cc
/  \                    /  \
*    *                  *    *
```

5. design the method `buildTree` for the classes that represent
binary search trees that consumes a list of `String`s and produces a binary search tree, with all `String`s in the list inserted in the tree. Verify that you produced the correct list by comparing the result of invoking the `inorder` method with the given list in sorted order.

## 5.8  Problem: Challenge - will not be graded

Design the method `delete` that deletes the given `String` from the binary search tree, while preserving the binary search tree property. If the tree does not contain the `String`, the method just returns the original tree.

## 5.9  Analytical Problem

The height of the binary tree is the longest path from the root `Node` to an `Empty Tree`. So for example, Tree-1 and Tree-2 in the first set of examples both have height 3, while the first tree in the Problem 5.7 has height 2.

1. What is the smallest and what is the greatest height of a binary serch tree with 63 nodes?, with 31 nodes?

2. Make examples of the binary search tree of minimum and maximum height with 15 nodes.

3. Show all possible binary search trees that contain the following
`Strings` and no others: aa bb cc dd