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Euclid discovered an algorithm that systematically and quickly reduces the size of the problem by replacing the original pair of numbers by smaller pairs until one of the pair becomes zero, at which point the GCD is the other number of the pair (the GCD of any number and 0 is that number) Here is Euclid s algorithm for finding the GCD of any two numbers A and B Repeat: If B is zero, the GCD is A.

Figure 11.3 shows a binary tree of size 10 and height 3. Node a is its root. The path from node h to node b has length 2. Node b is at level 1, and node h is at level 3. b is an ancestor of h, and h is a descendant of b. The part in the shaded region is a subtree of size 6 and height 2. Its root is node b.

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] More to do for the young woman in East St Louis, and thousands more like her, who has the grades, has the drive, has the will [he emphasizes the words and his slight pauses add power to the delivery], but doesn t have the money to go to college Now, don t get me wrong The people I meet in small towns and big cities, in diners and office parks, they don t expect government to solve all their problems They know they have to work hard to get ahead, and they want to [Obama stresses the words as he pinches his fingers to further accentuate his statement] Go into the collar counties around Chicago, and people will tell you they don t want their tax money wasted by a welfare agency or by the Pentagon.

To find all the binary trees of size 5, apply the recursive definition for binary trees. If t is a binary tree of size 5, then it must consist of a root node together with two subtrees the sum of whose sizes equals 4. There are four possibilities: The left subtree contains either 4, 3, 2, 1, or 0 nodes. First count all the binary trees of size 5 whose left subtree has size 4. From Example 11.3, we see that there are 14 different possibilities for that left subtree. But for each of those 14 choices, there are no other options because the right subtree must be empty. Therefore, there are 14 different binary trees of size 5 whose left subtree has size 4. Next, count all the binary trees of size 5 whose left subtree has size 3. From Example 11.2, we see that there are five different possibilities for that left subtree. But for each of those five choices, there are no

Otherwise: find the remainder R when dividing A by B replace the value of A with the value of B replace the value of B with the value of R For example, to find the GCD of 372 and 84, which we will show as: GCD(372, 84) Find GCD(84, 36) because 372/84 > remainder 36 Find GCD(36, 12) because 84/36 > remainder 12 Find GCD(12, 0) because 36/12 > remainder 0; Solved! GCD = 12 More formally, an algorithm is a sequence of computations that operates on some set of inputs and produces a result in a finite period of time In the example of the algorithm for designing stairs, the inputs are the total rise and total run The result is the best specification for the number of steps, and for the rise and run of each step.

other options because the right subtree must be a singleton. Therefore, there are five different binary trees of size 5 whose left subtree has size 3. Next, count all the binary trees of size 5 whose left subtree has size 2. There are only two different possibilities for that left subtree. But for each of those two choices, we have the same two different possibilities for the right subtree because it also must have size 2. Therefore, there are 2 2 = 4 different binary trees of size 5 whose left subtree has size 2. By similar reasoning, we find that there are five different binary trees of size 5 whose left subtree has size 1, and there are 14 different binary trees of size 5 whose left subtree has size 0. Therefore, the total number of different binary trees of size 5 is 14 + 5 + 4 + 5 + 14 = 42.

FULL BINARY TREES A binary tree is said to be full if all its leaves are at the same level and every interior node has two children. EXAMPLE 11.5 The Full Binary Tree of Height 3

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