87 lines
2.2 KiB
Markdown
87 lines
2.2 KiB
Markdown
|
||
# 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).
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|