## Overview

**2-3 and 2-3-4 trees.** Understand how to search and insert. Understand why
insertion technique leads to perfect balance.

**Terminology.** Know that a 2-3 tree is also called a B-Tree of order 3 and
that a 2-3-4 tree is a B-Tree of order 4. Know that an M-node is a node with M
children, e.g. a 4-node is a node containing 3 items and thus with 4 children.

**Performance of trees.** Tree height is between log_M N and log_2 N. All paths
are of the same height.

**1-1 correspondence between LLRBs and 2-3 trees.** Understand how to map a 2-3
tree to an LLRB and vice versa. You do not need to know how to perform tree
operations ‘natively’ on an LLRB.

**LLRBs.** BST such that no node has two red links touching it; perfect black
balance; red links lean left.

**LLRB get.** Exactly the same as regular BST get.

**LLRB performance.** Perfect black balance ensures worst case performance for
get and insert is ~ 2 log_2 N.

**LLRB insert.** You are not responsible for knowing how to insert into an LLRB
using rotations and color flipping. However, you should know how to convert back
and forth between 2-3 trees and LLRBs, and you should also know how to insert
into 2-3 trees. Thus, you should know effectively how one can insert into an
LLRB.

## Example Implementations

## Recommended Problems

### C level

- Draw the 2-3 tree that results when you insert the keys A B C D E F G in order.
- Draw a 2-3 tree with all 3 nodes. Why is the height log_3(N)? Draw the corresponding left-leaning red black tree (LLRB).
- How many compares does it take in the worst case to decide whether to go left, middle, or right from a 3 node?

### B level

- Draw the LLRB tree that results when you insert the keys A B C D E F G in order.
- Given an LLRB, is there exactly one corresponding 2-3 tree? Given a 2-3 tree, is there exactly one corresponding red-black tree?
- What is the worst case height of a 2-3-4-5 tree? Best case? Give your answers in tilde notation. Edit: An earlier
- For a 2-3 tree of height N, what is the best case number of compares to find a key? The worst case?
- What is the worst case height of an LLRB? Give your answer in tilde notation. Draw an example of a worst case tree.
- Problem 2 of my Spring 2013 midterm, part b only.
- Problem 5 of the Fall 2014 midterm.

### A level

- Derive a simple procedure for deleting from a 2-3 tree. Your procedure must take $O(\log N)$ time.
- In class, we always chose the left-middle object when splitting a 4 node. Suppose we used the right-middle object instead. Would the idea of a 2-3-4 tree still make sense? Are there any advantages or disdvantages?
- In class, we always chose the left-middle object when splitting a 4 node. Suppose we used the left-most object instead. Would the idea of a 2-3-4 tree still make sense? Are there any advantages or disdvantages?
- Show that in the worst case, we need to perform at most log N LLRB operations (color flips and rotates) after an insert. Give an example of a tree and an item X such that insertion of X requires log N LLRB operations.