# Lecture 21 --- Trees, Part IV

Review from Lecture 19
- Breadth-first and depth-first tree search
- Increement/decrement operator
- Tree height, longest-shortest paths, breadth-first search
- Last piece of ds_set: removing an item, erase
- Erase with parent pointers, increment operation on iterators
- Limitations of our ds set implementatioN

## Today’s Lecture

- Red Black Trees
- B+ Trees

## 21.1 Red-Black Trees
In addition to the binary search tree properties, the following 
red-black tree properties are maintained throughout all 
modifications to the data structure:

- Each node is either red or black.
- The NULL child pointers are black.
- Both children of every red node are black.
- Thus, the parent of a red node must also be black.
- All paths from a particular node to a NULL child pointer contain the same
  number of black nodes.


What tree does our **ds_set** implementation produce if we insert the
numbers 1-14 **in order**?

The tree at the right is the result using a red-black tree.  Notice how the tree is still quite balanced.  

Visit these links for an animation of the sequential insertion and re-balancing:

http://babbage.clarku.edu/~achou/cs160fall03/examples/bst_animation/RedBlackTree-Example.html

https://www.cs.usfca.edu/~galles/visualization/RedBlack.html

http://www.youtube.com/watch?v=vDHFF4wjWYU&noredirect=1

What is the best/average/worst case height of a red-black tree with $n$ nodes?

 
 
 
 
 
 

What is the best/average/worst case shortest-path from root to leaf node in a red-black tree with $n$ nodes?

 
 
 
 
 
 

## Exercise 21.2
Fill in the tree on the right with the integers 1-7 to make a binary search tree.  Also, color each node "red" or "black" so that the tree also fulfills the requirements of a Red-Black tree. 



- Draw a diagram of a possible memory layout for a ds set containing the numbers 16, 2, 8, 11, and 5. Is there
only one valid memory layout for this data as a ds set? Why?

 
 
 
 
 
 

- In what order should a forward iterator visit the data? Draw an abstract table representation of this data
(omits details of TreeNode memory layout).