From 38bacc286094f1f60dd74476940097aab4cf4874 Mon Sep 17 00:00:00 2001 From: Jidong Xiao Date: Thu, 27 Mar 2025 20:42:23 -0400 Subject: [PATCH] adding rb trees --- lectures/rb_trees/BplusTrees.png | Bin 0 -> 25576 bytes lectures/rb_trees/README.md | 129 +++++++++++++++++++++++++ lectures/rb_trees/Red_Black.png | Bin 0 -> 92140 bytes lectures/rb_trees/Red_Black_fillin.png | Bin 0 -> 76284 bytes lectures/rb_trees/Trinary_trees.png | Bin 0 -> 57112 bytes 5 files changed, 129 insertions(+) create mode 100644 lectures/rb_trees/BplusTrees.png create mode 100644 lectures/rb_trees/README.md create mode 100644 lectures/rb_trees/Red_Black.png create mode 100644 lectures/rb_trees/Red_Black_fillin.png create mode 100644 lectures/rb_trees/Trinary_trees.png diff --git a/lectures/rb_trees/BplusTrees.png b/lectures/rb_trees/BplusTrees.png new file mode 100644 index 0000000000000000000000000000000000000000..72bdfb02edbc8c6a7add626cc445627651a21af0 GIT binary patch literal 25576 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z*FpPQ<_NWHrV#FTFV9{a?PWk7^guF$?8`pC;k7ow@dSL?x3 z{4BOxd+NJaf&lb<0_ZuJN#)m-AjlW^=dwF_H_P}xDWm=Na8q=n^yKdr<-a|FdKv0n K(tV}v82Mkm)BYU* literal 0 HcmV?d00001 diff --git a/lectures/rb_trees/README.md b/lectures/rb_trees/README.md new file mode 100644 index 0000000..cf248aa --- /dev/null +++ b/lectures/rb_trees/README.md @@ -0,0 +1,129 @@ + +# Lecture 21 --- Trees, Part IV + +Review from Lecture 20 +- 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 root node is always 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. + +![alt text](Red_Black.png "Red_Black Tree example") + +What tree does our **ds_set** implementation produce if we insert the +numbers 1-14 **in order**? +  +  +  +  +  +  + +The tree at the top 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. + +![alt text](Red_Black_fillin.png "Red_Black Tree Fill In example") + +Which nodes are red? + +**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. + +## 21.3 Trinary Tree + +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 +the examples below, the tree on the left will return true and the tree on the right will return false. + +```cpp +class Node { + public: + int value; + Node* left; + Node* middle; + Node* right; + }; +``` +![alt text](Trinary_trees.png "Trinary Trees example") + +## 21.4 B+ Trees + +Unlike binary search trees, nodes in B+ trees (and their predecessor, the B tree) have up to b children. Thus +B+ trees are very flat and very wide. This is good when it is very expensive to move from one node to another. +- B+ trees are supposed to be associative (i.e. they have key-value pairs), but we will just focus on the keys. +- Just like STL map and STL set, these keys and values can be any type, but keys must have an operator< +defined. +- In a B tree key-value pairs can show up anywhere in the tree, in a B+ tree all the key-value pairs are in the +leaves and the non-leaf nodes contain duplicates of some keys. +- In either type of tree, all leaves are the same distance from the root. +- The keys are always sorted in a B/B+ tree node, and there are up to b − 1 of them. They act like b − 1 binary +search tree nodes mashed together. +- In fact, with the exception of the root, nodes will always have between roughly b/2 and b − 1 keys (in our +implementation). +- If a B+ tree node has k keys key0, key1, key2, . . . , keyk−1, it will have k + 1 children. The keys in the leftmost +child must be < key0, the next child must have keys such that they are ≥key0 and < key1, and so on up to +the rightmost child which has only keys ≥keyk−1. + +A B+ tree visualization can be seen at: https://www.cs.usfca.edu/~galles/visualization/BPlusTree.html + +![alt text](BplusTrees.png "Bplus_Trees example") + +Considerations in a full implementation: +- What happens when we want to add a key to a node that's already full? +- How do we remove values from a node? +- How do we ensure the tree stays balanced? +- How to keep laves linked together? WHy would we want this? +- How to represent key-value pairs? + +## 21.5 Exercise + +Draw a B+ tree with b=3 with values inserted in the order 1,2,3,4,5,6. Now draw a B+ tree with b=3 and values inserted in the order 6,5,4,3,2,1. 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