Thinking in Pictures

Graphs

Let S be a set and R: S x S be a relation on S.
The graph G = (S,R) is a picture of the relation.

Example: X = {a,b}, S = powerset(X), R = {[x,y] | x is a proper subset of y}

Each member of S is a node in the graph, represented by a dot.
Each member of R is an arc (or an edge), represented by an arrow.

An arrow, x -> y, represents an element [x,y] in R:
y is called the head of the arrow (as in arrowhead)
and x is called the tail of the arrow

In our example, one of the edges (arcs) is the arrow {} -> {a}

The types of graphs that we will be be using are directed graphs .
We can identify a graph as being a directed graph because the arcs are arrows , and not just lines ; directed means that we have to follow the directions of the arrows .

Definitions

• An initial node is a node we leave when traversing the arc.
• A terminal node is a node we arrive at when traversing the arc.
• The in-degree of a node n is the # arcs whose heads point to node n
• The out-degree of a node n is the # arcs whose tails start at node n
• A path is a sequence of edges [x_1,y_1], [x_2,y_2], ..., [x_n,y_n] such that for 1 < i <= n : y_i-1 = x_i
  Ex: [{},{a}],[{a},{ab}] is a path
• A cycle is a path such that x_1 = y_n (also called a circuit) A cycle is called simple if it does not contain a cyclic subpath.
• A graph is called acyclic if it has no cycles
• A labelled directed graph is a directed graph in which each arc is assigned a label or a value.
  Ex: Airline fares from point A to point B: Not only is this a relation, but we can assign an airfare for each "leg" of the flight. Notice, then, that A->B and B->A can be two arcs with different air fares (labels).

Trees

1. has no (directed) cycles, i.e., it is acyclic.
2. has one single node, called the root of the tree, which has in-degree 0
3. all other nodes have in-degree 1
4. there is a path from the root to every node

1. The path from the root to a given node is unique
2. The #arcs = #nodes-1
   (Hmmm, This seems like something we could prove using Induction!!)
3. The subset of nodes that have out-degree 0 are called the leaf nodes.

Proving Properties of Trees

To prove properties about trees (like #2 in the "Other properties..." above, we need a Recursive Definition of a Tree. In Computer Science, Binary Trees are typically the ones we prove properties about, because they are the most prevalent structures, both theoretically and in practice (searching and sorting algorithms).

(Recursive) Definition: Binary Tree

1. (Basis) T is empty
2. (Recursion) T contains a distinguished node r, called the root of T, and the other nodes (T-{r}) are divided into two disjoint sets, called the left subtree of r and the right subtree of r , each of which in itself is a Binary Tree.

• Your book gives a more explicit definition of a tree, from that of a graph, by explicitly naming the root r, of the tree: T=(S,A,r). A is an adjacency relation, which says that for any [x,y] in A it is said that x is adjacent to y, Adjacent to means there is an arc from x to y, i.e., x->y, in the tree.
• If x->y then y is child of x, and x is the parent of y.
• The depth of a node is defined recursively: The depth of the root is zero; the depth of any other node is the depth of it's parent plus 1.
• The height of a tree is the maximum of the depths of all nodes in the tree.
• y is a descendent of x if there is a path from x to y.
  x is an ancestor of y if there is a path from s to y.
  Note: that descendent is the transitive of child and ancestor the transitive of parent .
• We will assume that a tree is an ordered graph, which means that the children (or siblings) of a node have a left-to-right order.
• The frontier of a tree is an ordered list of leaf nodes, generated from left to right.
• Given two nodes x and y, the minimal common ancestor of x and y is a node z, where z is an ancestor of x and y, and z is also the ancestor that is the closest ancestor to x and y.