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+# 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?
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+What is the best/average/worst case shortest-path from root to leaf node in a red-black tree with $n$ nodes?
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+## 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. 
+
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+- 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?
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+- In what order should a forward iterator visit the data? Draw an abstract table representation of this data
+(omits details of TreeNode memory layout).
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