ECS!!) Notes for Monday, October 30, 1995

ECS110 Lecture Notes for Monday, October 30'th 1995

Professor Rogaway Conducting

Lecture 13

Scribe of the day : Arash Zolfaghari

Reminder:

Midterm #1 & Programing Assignment #1 Will be returned Wednsday.
Try the EXERCISE mentioned in the class regarding Traversal of a Binary Tree.

Previous Lecture Topic:

Lists

Next Lecture Topic:

Illustrations of Today's Topic using algorithms

Today:

Graph Theory Terminology
Trees:
1. General Trees
  - Defining a Tree
  - Rooted Trees
  - Leaf
  - Internal Node
  - Level
  - Subtree
  - Ordered Tree
2. Binary Trees
  - Defining a Binary Tree
  - Complete Tree
  - Relationship Between Depth and number of Nodes
  - Relationship Between Number of Leafs and Missing Nodes
  - Representation of Binary Trees
  - As Arrays
  - As Linked Lists
  - The efficiency Factor
    - For a complete Tree
    - For an incomplete Tree
  - Ease of Use
  - Translating Any Ordered Forest into Binary Tree form
  - Going inside a Binary Tree

Graph Theory Terminology
1. Defining Graph
  graph G is a non empty, finite set of vertecies V and a set of 2 element subset of V denoted by E called nodes.
  G=(V,E)
2. Adjacent Members
  Two Vertices are Adjacent when they share an edge.
3. Degree of Vertex V
  This is the number of vertices that are adjacent to this node.
4. Path
  a path in G is a sequence (V1,..........Vn) such that (V1,V2),(V2,V3), ..... , (Vn-1,Vn) are all edges in the set E.
5. Being Connected
  When every 2 vertices lie on the same common path, they are said to be connected. An example of a connected Graph is a Tree.
6. Cycle
  A cycle is a path from (V1.....Vn) where V1=Vn for n>3.
7. Acyclical
  A graph is acyclic if none of the paths traveled return back to the first vertex. An example of an Acyclic graph is a Tree.
Trees:
1. General Trees
  
  - Defining a Tree
  A tree is a iconnected, acyclic graph
  A forest is a set of trees
  
  - Rooted Trees
  This type of tree, also known as Oriented Tree, is a tree where one vertex of which is distinguished as - The Root. This root is usually placed at the top of the tree. A rooted tree can therefore be even represented using arrows extending from the root as can be seen in Figure 5. Also in Figure six, both trees represent the same thing, a root with a degree of 5. T=(V,E,r) where r, the root, is an element of V (meaning root is the vertex)
  
  - Leaf
  This is defined as a node which has a degree of one. In other words, this node bears no children. See Figure 7.
  
  - Internal Node
  This is a node with a degree of n =>2, found between edges.
  
  - Level
  The root is assigned level 1. The level is the depth of the branch division. (Like generations).
  
  - Subtree
  In describing trees we describe branches going out from a vertex as a subtree. In particular in Rooted Trees we refer to them as the children where the parent is the vertex where the links are extending from.
  
  - Ordered Tree
  This is an oriented tree where the siblings are numbered 1, 2, .... , n in the ascending order of age (where the left hand side sibling is the youngest and the right hand side sibling is the oldest. So this is a method of organizing our tree in a way that siblings are easily distinguished to be younger or older than the other.
2. Binary Trees
  
  - Defining a Binary Tree
  A finite set of nodes which is either empty, or has a distinguished node called the the Root, and two disjoint binary trees: A Left binary subtree , a Right binary subtree. A child in the left subtree is not the same as having the child in the right subtree as illustrated by Figure 11.
  
  - Complete Tree
  A tree is complete when every level i (except the last) has all 2^i nodes at level i filled from Left to Right.
  
  - Relationship Between Depth and number of Nodes
  A binary tree of depth d has at most (2^d)-1 nodes.
  The Depth of a COMPLETE tree is the upper limit of log(n+1) where the base of the log is 2.
  
  - Relationship Between Number of Leafs and Missing Nodes
  Number of Missing leafs == Number of Nodes +1
  - Representation of Binary Trees
  - As Arrays
    Each box with in the array (starting with 1 as the root), can be algorithmically given the appropiate nodes. Let's use this algorithm as a general way of placing the member with in an array: Given a node i :
    Parent(i) = Lower Limit of (i/2)
    Left Child(i) = 2*i RightChild(i) = (2*i)+1
  - As Linked Lists
    Abstractly Speaking, the Link list members will contain important informations:
    
    Pointer to the Left Child
    Pointer to the Right Child
    What value is Stored Here.
  - The efficiency Factor for an array representation
    
    - For a complete Tree
    Uses just enough space to fill up its array (hence the term complete!). Traversal to a child or parent is easily done algorithmically.
    
    - For an incomplete Tree
    Excessive space can be wasted this way if too many parents do not have complete set of children.
  - Ease of Use
    Arrays tend to be easier to use since we can find where a child or a parent is by simply calculating it. In an array we also have the added advantage of knowing where the root parent is at all time (for the ease of use, we use box 1 as root). On the other hand, a linked list, if not explicitly stated, needs explicit addressing to a child who wants to go back to a parent (or a root).
  - Translating Any Ordered Forest into Binary Tree form
  This can be used to translate forests into binary tree format. Traversing down the tree, form a new binary tree by the following criteria:
  
  Use the Left node of the new binary tree member to save the location of the youngest child from the forest.
  Use the Right node of the new binary tree member to save the location of the older sibling from the forest.
  - traveling through a Binary Tree
  - Traversing
    Traveling through the binary tree, this is to visit each node EXACTLY once
  - Exploring
    This is traveling a tree and visiting each node at least Once
  - EXCERCISE Traversal
    Write a procedure for traversing the maze below: