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lectures/18_trees_I/README.md

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+# Lecture 17 --- Trees, Part I
+
+## Review from Lecture 16
+
+- STL set container class (like STL map, but without the pairs!)
+- set iterators, insert, erase, find
+
+## Today’s Lecture
+
+- Binary trees, binary search trees
+- Implementation of ds_set class using binary search trees
+- In-order, pre-order, and post-order traversal
+
+## Overview: Lists vs Trees vs Graphs
+- Trees create a hierarchical organization of data, rather than the linear organization in linked lists (and arrays and vectors).
+- Binary search trees are the mechanism underlying maps & sets (and multimaps & multisets).
+- Mathematically speaking: A _graph_ is a set of vertices connected by edges. And a tree is a special graph that has no _cycles_. The edges that connect nodes in trees and graphs may be _directed_ or _undirected_.
+
+## 17.1 Definition: Binary Trees
+
+- A binary tree (strictly speaking, a “rooted binary
+tree”) is either empty or is a node that has
+pointers to two binary trees.
+- Here’s a picture of a binary tree storing integer
+values. In this figure, each large box indicates a
+tree node, with the top rectangle representing the
+value stored and the two lower boxes representing
+pointers. Pointers that are null are shown with a
+slash through the box.
+
+![alt text](binary_tree.png "binary tree")
+
+- The topmost node in the tree is called the root.
+- The pointers from each node are called left and
+right. The nodes they point to are referred to as
+that node’s (left and right) children.
+- The (sub)trees pointed to by the left and right
+pointers at any node are called the left subtree
+and right subtree of that node.
+- A node where both children pointers are null is
+called a leaf node.
+- A node’s parent is the unique node that points to
+it. Only the root has no parent.
+
+## 17.2 Definition: Binary Search Trees
+
+- A binary search tree (often abbreviated to
+BST) is a binary tree where at each node
+of the tree, the value stored at the node is
+  – greater than or equal to all values
+stored in the left subtree, and
+
+  – less than or equal to all values stored in
+the right subtree.
+
+- Here is a picture of a binary search tree
+storing string values.
+
+![alt text](bst.png "binary search tree")
+
+## 17.3 Definition: Balanced Trees
+
+- The number of nodes on each subtree of each node in a
+“balanced” tree is approximately the same. In order to
+be an exactly balanced binary tree, what must be true
+about the number of nodes in the tree?
+- In order to claim the performance advantages of trees, we must assume and ensure that our data structure
+remains approximately balanced. (You’ll see much more of this in Intro to Algorithms!)
+
+## 17.4 Exercise
+
+Consider the following values:
+4.5, 9.8, 3.5, 13.6, 19.2, 7.4, 11.7
+
+1. Draw a binary tree with these values that is NOT a binary search tree.
+
+2. Draw two different binary search trees with these values. Important note: This shows that the binary search
+tree structure for a given set of values is not unique!
+
+3. How many exactly balanced binary search trees exist with these numbers? How many exactly balanced
+binary trees exist with these numbers?
+
+## 17.5 Beginning our implementation of ds_set: The Tree Node Class
+
+- Here is the class definition for nodes in the tree. We will use this for the tree manipulation code we write.
+
+```cpp
+template <class T> class TreeNode {
+public:
+	TreeNode() : left(NULL), right(NULL) {}
+	TreeNode(const T& init) : value(init), left(NULL), right(NULL) {}
+	T value;
+	TreeNode* left;
+	TreeNode* right;
+};
+```
+
+- Note: Sometimes a 3rd pointer — to the parent TreeNode — is added.
+
+  ![alt text](ds_set_diagram.png "ds set diagram")
+
+## 17.6 Exercises
+
+1. Write a templated function to find the smallest value stored in a binary search tree whose root node is pointed
+to by p.
+
+2. Write a function to count the number of odd numbers stored in a binary tree (not necessarily a binary search
+tree) of integers. The function should accept a TreeNode<int> pointer as its sole argument and return an
+integer. Hint: think recursively!
+
+## 17.7 ds_set and Binary Search Tree Implementation
+
+- A partial implementation of a set using a binary search tree is provided in this [ds_set_starter.h](ds_set_starter.h). We will continue to study this implementation in Lab 10 & the next lecture.
+- The increment and decrement operations for iterators have been omitted from this implementation. Next week
+in lecture we will discuss a couple strategies for adding these operations.
+- We will use this as the basis both for understanding an initial selection of tree algorithms and for thinking
+about how standard library sets really work.
+
+## 17.8 ds_set: Class Overview
+
+- There is two auxiliary classes, TreeNode and tree_iterator. All three classes are templated.
+- The only member variables of the ds_set class are the root and the size (number of tree nodes).
+- The iterator class is declared internally, and is effectively a wrapper on the TreeNode pointers.
+  – Note that operator* returns a const reference because the keys can’t change.
+
+  – The increment and decrement operators are missing (we’ll fill this in next week in lecture!).
+
+- The main public member functions just call a private (and often recursive) member function (passing the root
+node) that does all of the work.
+- Because the class stores and manages dynamically allocated memory, a copy constructor, operator=, and
+destructor must be provided.
+
+## 17.9 Exercises
+
+1. Provide the implementation of the member function ds_set<T>::begin. This is essentially the problem of
+finding the node in the tree that stores the smallest value.
+
+
+
+
+2. Write a recursive version of the function find.
+
+## 17.10 In-order, Pre-Order, Post-Order Traversal
+
+- One of the fundamental tree operations is “traversing” the nodes in the tree and doing something at each node.
+The “doing something”, which is often just printing, is referred to generically as “visiting” the node.
+- There are three general orders in which binary trees are traversed: pre-order, in-order and post-order.
+ In order to explain these, let’s first draw an “exactly balanced” binary search tree with the elements 1-7:
+
+  – What is the in-order traversal of this tree? Hint: it is monotonically increasing, which is always true for
+an in-order traversal of a binary search tree!
+
+
+
+  – What is the post-order traversal of this tree? Hint, it ends with “4” and the 3rd element printed is “2”.
+
+
+
+  – What is the pre-order traversal of this tree? Hint, the last element is the same as the last element of the
+in-order traversal (but that is not true in general! why not?)
+