87 lines
2.7 KiB
Markdown
87 lines
2.7 KiB
Markdown
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# Lecture 21 --- Trees, Part IV
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Review from Lecture 19
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- Breadth-first and depth-first tree search
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- Increement/decrement operator
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- Tree height, longest-shortest paths, breadth-first search
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- Last piece of ds_set: removing an item, erase
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- Erase with parent pointers, increment operation on iterators
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- Limitations of our ds set implementatioN
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## Today’s Lecture
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- Red Black Trees
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- B+ Trees
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## 21.1 Red-Black Trees
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In addition to the binary search tree properties, the following
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red-black tree properties are maintained throughout all
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modifications to the data structure:
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- Each node is either red or black.
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- The NULL child pointers are black.
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- Both children of every red node are black.
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- Thus, the parent of a red node must also be black.
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- All paths from a particular node to a NULL child pointer contain the same
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number of black nodes.
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What tree does our **ds_set** implementation produce if we insert the
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numbers 1-14 **in order**?
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The tree at the right is the result using a red-black tree. Notice how the tree is still quite balanced.
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Visit these links for an animation of the sequential insertion and re-balancing:
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http://babbage.clarku.edu/~achou/cs160fall03/examples/bst_animation/RedBlackTree-Example.html
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https://www.cs.usfca.edu/~galles/visualization/RedBlack.html
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http://www.youtube.com/watch?v=vDHFF4wjWYU&noredirect=1
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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
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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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Which nodes are red?
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**Note:** Red-Black Trees are just one algorithm for **self-balancing binary search tress**. We have many more, including the AVL trees that we discussed last week.
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## Trinary Tree
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A **trinary tree** is similar to a binary tree except that each node has at most 3 children. Write a **recursive** function named **EqualsChildrenSum** that takes one argument, a pointer to the root of a trinary tree, and returns true if the value at each non-leaf node is the sum of the values of all of its children and false otherwise. In
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the examples below, the tree on the left will return true and the tree on the right will return false.
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```cpp
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class Node {
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public:
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int value;
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Node* left;
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Node* middle;
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Node* right;
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};
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```
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