Material covered through next lecture (on Wednesday) will be on Midterm.

This week’s resitation sessions will be midterm review.

Next Monday will be Midterm review. Do not miss this.

BinarySearchTree Class

Look up and understand this code, it’s provided in the book PDF.

Comparable<? super AnyType>says it has to be comparable to itself or any superclass of itself.

He says it’s gratuitous having two Constructors since when you insert, you replace the null reference with a single leaf node.

public boolean contains(AnyType x) Method

Returns contains(x, root). The true contains method has to take in the root. This one does not and is not recursive, it actually calls the private recursive method with root.

We see this public / private pairing a lot in this.

Insert Method

public boolean contains(AnyType x)

The method takes in a node we want to insert. If it’s already in the tree, we don’t do anything. If it’s not, we add it in, and make sure the rules of the tree are upheld.

public void insert(AnyType x) {
    // We assign to root here just in case the tree is empty.
    // So the recursive `insert` that this calls returns the "new"
    // root of the tree
    root = insert(x, root);
}

The private method is recursive and similar to contains in its logic.

Find Min and Find Max

We basically just search left (right) until we find a left null reference and return that node with the null pointer.

These are tail recursions; when you make a recursive call as the very last line of your method. You’re effectively iterating across the dataset. These can very easily be replace with while loops.

In the text, findMax is written iteratively as a while loop instead of as a recursion.

Height Method

Find the longer of the left and right subtrees (recursively) and add one (the root).

Print Tree

Standard in-order traversal. Will print the data in sorted order.

Remove

Leaf nodes are simple. Just drop them.

What if we delete the root?

• Replace with the smallest thing on the right hand side.
• Or the largest thing on the left hand side.

So the steps are:

1. Find the largest on the left side using findMax
2. Replace root with it
3. Recursively remove it

public method very similar to insert method.
private method starts out with base case similar to others (null tree).

Pretty simple if the node just has a single child. If it has two, it’s kinda hard.

Look over the code and understand this. Fading in class fast.

Lazy Deletion

Add a field to the TreeNode class valid. If it’s true, the node is marked to exist. If it’s false, the node is marked to not exist. So in lazy deletion, you just do a contains check and if it finds it, you just flip the value to false.

All the other methods need to be adjusted to take account of this though and check whether nodes are valid or not.

findMax and findMin actually become quite tough.

AVL Trees

Binary search trees with an added condition: the height of the left subtree cannot differ from the height of the right subtree by more than one. If at any time, an insertion or deletion is made that disrupts this balance, you need to perform a single or double rotation to fix it.