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In the example of finding the GCD of two numbers, the inputs are the two numbers, and the result is the GCD Often there are several ways to solve a class of problems, several algorithms that will get the job done The question then is which algorithm is best In the case of algorithms for computing, computer scientists have developed techniques for analyzing the performance and judging the relative quality of different algorithms REPRESENTING ALGORITHMS WITH PSEUDOCODE In computer science, algorithms are usually represented as pseudocode Pseudocode is close enough to a real programming language that it can represent the tasks the computer must perform in executing the algorithm Pseudocode is also independent of any particular language, and uncluttered by details of syntax, which characteristics make it attractive for conveying to humans the essential operations of an algorithm..

The tree shown in Figure 11.6 is the full binary tree of height 3. Note that it has 15 nodes: 7 interior nodes and 8 leaves.

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Theorem 111 The full binary tree of height h has l = 2h leaves and m = 2h 1 internal nodes Proof: The full binary tree of height h = 0 is a single leaf node, so it has n = 1 node, which is a leaf Therefore, since 2h 1 = 20 1 = 1 1 = 0, and 2h = 20 = 1, the formulas are correct for the case where h = 0 More generally, let h > 0 and assume (the inductive hypothesis) that the formulas are true for all full binary trees of height less than h Then consider a full binary tree of height h Each of its subtrees has height h 1, so we apply the formulas to them: lL = lR = 2h 1 and mL = mR = 2h 1 1.

insure reliable communication over the inherently unreliable IP network layer The end-to-end (transport) layer protocol performs this magic every day END-TO-END (TRANSPORT) LAYER PROTOCOL The transport layer protocol of the internet is transmission control protocol (TCP) TCP uses the IP layer to send messages through the internet, and TCP adds a whole set of services that together insure that complete messages always arrive, uncorrupted, in order, without duplication of data, at the intended destination, regardless of hardware or network failures or changes during the time the many datagrams comprising the message are being sent TCP is a connection-oriented protocol, which means that the two computers first establish a connection with one another before either begins sending data To do this, one computer, the server, reserves a port for communication.

(These are the number of leaves in the left subtree, the number of leaves in the right subtree, the number of internal nodes in the left subtree, and the number of internal nodes in the right subtree, respectively) Then l = lL + lR = 2h 1 + 2h 1 = 2 2h 1 = 2h and m = mL + mR + 1 = (2h 1 1) + (2h 1 1) + 1 = 2 2h 1 1 = 2I 1 Therefore, by the (Second) Principle of Mathematical Induction, the formulas must be true for full binary trees of any height h 0 By simply adding the formulas for m and l, we obtain the first corollary Corollary 111 The full binary tree of height h has a total of n = 2h+1 1 nodes.

Some writers call this the passive open of a connection, for nothing happens outside of the server; the server simply makes itself ready to communicate When another computer, the client, wishes to communicate with the server, the client contacts the server requesting a connection This step is sometimes called the active open To do so, the client must identify the server, and must know in advance the port on which the server is listening Since each computer has thousands of ports available, you may be wondering how the typical client application could possibly guess on which port to call the server The answer is that the use of a small number of ports is standardized, and those port numbers are well known.

By solving the formula n = 2h+1 1 for h, we obtain this corollary: Corollary 112 The full binary tree with n nodes has height h = lg(n+1) 1 Note that the formula in Corollary 112 is correct even in the special case where n = 0: The empty binary tree has height h = lg(n+1) 1 = lg(0+1) 1 = lg(1) 1 = 0 1 = 1..

CHAP. 11]

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