programming-for-dummies

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Creating Trees
A related problem, dubbed the Traveling Salesman problem, takes this one
step further by finding the shortest round-trip route through a graph where
the same node is both the starting and ending point. In this case, the shortest
route through a graph may not necessarily be the shortest round-trip
route to return to the same starting point.
Both the Traveling Salesman and the Chinese Postman problem can get
more complicated with a weighted or directed graph. A directed graph may
restrict movement in one direction, such as traveling through one-way
streets in a city, whereas a weighted graph can make one route longer than
two shorter routes combined. Adding in directed and weighted graphs can
alter the best solution.
If you ever looked up directions on a map Web site such as MapQuest,
you’ve used a graph to find the most efficient way from one location to
another.
Connecting nodes in a graph
Another use for graphs involves topological graph theory. This problem is
highlighted by the Three Cottage problem with three cottages needing to
connect to the gas, water, and electricity companies, but their lines can’t
cross each other. (It’s impossible, by the way.)
Connecting lines in a graph without crossing is a problem that circuit board
designers face in the placement of chips. Another example of eliminating
intersections involves transportation designs, such as the design of highways
or railroad tracks.
Creating Trees
Graphs typically represent a chaotic arrangement of data with little or no
structure. To give graphs some form of organization, computer scientists
have created special graphs dubbed trees. Like a graph, a tree consists of
nodes and edges, but unlike a graph, a tree organizes data in a hierarchy, as
shown in Figure 5-5.
A tree arranges a graph in a hierarchy with a single node appearing at the
top (or the root node) and additional nodes appearing connected underneath
(or internal nodes). If a node has no additional nodes connected underneath,
those nodes are leaf nodes, as shown in Figure 5-6.