CS 300 Final Exam Fall 2006 Jeremy Johson Name: Instructions: The exam is due by Fri. Dec. 15 at 11:59pm. It should be submitted via webct and if it is not received on time, it will not be accepted. The exam must be completed independently - you may use class notes and worksheets, and other sources, but you may not consult with anyone. By submitting this exam you verify that you have completed the exam yourself. If you do not understand a question, you may send me email and I will try to clarify the problem. Make sure you fill in your name in the top of the worksheet. There are three questions worth 30, 30, and 40 points.

In assignment 4, you were asked to write a recursive procedure that converted a Forest to a Binary Tree. In this question, you will write a recursive procedure that converts a Binary Tree to a Forest. The posted solution to assignment 4 provides a solution to the Forest to Binary Tree problem (you will be required - see below - to use the same data structure used in the solution to assignment 4). Make sure that you test your code (e.g. convert a binary tree to a forest and then back again). You should represent forests as a list of trees, and an aribtrary tree as a list of length 2 [value, children], where children is a list of trees. Binary trees should be represented as a list of length 3 [value, left, right], where left and right are lists representing binary trees. In all three cases, the empty forest, empty tree, and empty binary tree is represented by the empty list.

In the combstruct lecture from class, the following grammar from unordered trees was presented. Recall that an unordered tree can have an arbitrary number of children but it does not distinguish the order in which the children occur. There are 4 unordered trees of size 4: * * * * / \ | | / | \ * * * * * * * / | / \ * * * * | * Using the command gfseries from the combstruct package, compute the number of unordered trees of size 0 through 9. restart; with(combstruct);

Ny5JKGFnZmVxbnNHNiJJL2FnZm1vbWVudHNvbHZlR0YkSSphZ2ZzZXJpZXNHRiRJK2FsbHN0cnVjdHNHRiRJJmNvdW50R0YkSSVkcmF3R0YkSSlmaW5pc2hlZEdGJEknZ2ZlcW5zR0YkSSlnZnNlcmllc0dGJEkoZ2Zzb2x2ZUdGJEksaXRlcnN0cnVjdHNHRiRJK25leHRzdHJ1Y3RHRiQ=

unorderedtree := { T = Prod(Z,Set(T)) };

PCMvSSJURzYiLUklUHJvZEdGJTYkSSJaR0YlLUkkU2V0R0YlNiNGJA==

A free tree of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=is a connected graph with 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edges. A free tree does not distinguish a root node from any of the other nodes. Of the four unordered trees above, there are only two distinct free trees as the first two and the last two are not distinguishable as free trees. Given the generating function A(z) for unordered trees, The generating function, F(z)=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 for free trees (i.e. 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is the number of free trees with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=nodes) satisfies satisfies the equation 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 To understand this notation, suppose 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 then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZGLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUklbXN1cEdGJDYlLUYsNiVRInpGJ0YvRjItRiM2Iy1JI21uR0YkNiRRIjJGJy9GM1Enbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkYtSSNtb0dGJDYwUSI9RidGRi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGUS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRl1vLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUZDNiRGYm9GRi1GTDYwUSJ+RidGRkZPRlJGVEZWRlhGWkZmbi9GaW5RIUYnL0Zcb1EkMGVtRicvRl9vRl5wRmBvRmNvLUZMNjBRIitGJ0ZGRk9GUkZURlZGWEZaRmZuRmhuL0Zcb1EwbWVkaXVtbWF0aHNwYWNlRicvRl9vRmRwRmBvRmNvRmhvRkItRkw2MFEnJnNkb3Q7RidGRkZPRlJGVEZWRlhGWkZmbkZobkZdcEZfcEZgb0Zjb0Y6RmBwLUZDNiRRIjNGJ0ZGRmZwLUY7NiVGNUZARkhGYHAtRkM2JFEiN0YnRkZGZnAtRjs2JUY1LUYjNiNGaXBGSEZLRmZvRmhvRmBwRmhvRkJGZnBGOkZgcEZpcEZmcC1GOzYlRj0tRiM2Iy1GQzYkUSI0RidGRkZIRmBwRl5xRmZwLUY7NiVGPS1GIzYjLUZDNiRRIjZGJ0ZGRkgtRkw2MFEiLEYnRkZGTy9GU0YxRlRGVkZYRlpGZm5GaG5GXXAvRl9vUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGJ0Zgb0Zjb0Zobw==and 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 Using the first 10 terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYn, that you just generated, and this formula, compute as many terms in the series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlcvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=as you can. How many of them are the same as what would be obtained if the entire series for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRInpGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEifkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRlcvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=was used?

Let 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEicUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRI25rRidGMkY1LyUvc3Vic2NyaXB0c2hpZnRHUSIwRic= is the probability that the path length of a random binary search tree (BST) of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RidGNQ==is equal to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIi5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GNjYwUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSFGJ0ZNRlBGUkZVRlg=The probability generating function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYvNiVRInVGJ0YyRjUvRjZRJ25vcm1hbEYn satisfies the recurrence: 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 with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUkjbW5HRiQ2JFEiMEYnL0Y2USdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRi82JVEidUYnRjJGNUY+LUkjbW9HRiQ2MFEiPUYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zdby8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GOzYkUSMxLkYnRj4tRkw2MFEifkYnRj5GT0ZSRlRGVkZYRlpGZm4vRmluUSFGJy9GXG9RJDBlbUYnL0Zfb0ZfcEZgb0Zjb0Zpbw==Use this recurrence to write a recursive Maple procedure to compute 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 use it to compute 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You should expand and sort 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 see the possible path lengths (i.e. the degrees of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEidUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==) and probabilities (i.e. the probability that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEia0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=is the path length of a tree of size LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric= is 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 coefficient of 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 Verify that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEiUUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYjLUYvNiVRIm5GJ0YyRjUvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUkjbW5HRiQ2JFEiMUYnL0Y2USdub3JtYWxGJ0ZJLUkjbW9HRiQ2MFEiPUYnRkkvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRlEvJSlzdHJldGNoeUdGUS8lKnN5bW1ldHJpY0dGUS8lKGxhcmdlb3BHRlEvJS5tb3ZhYmxlbGltaXRzR0ZRLyUnYWNjZW50R0ZRLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zdby8lKG1pbnNpemVHRkgvJShtYXhzaXplR1EpaW5maW5pdHlGJ0ZFLUZMNjBRIn5GJ0ZJRk9GUkZURlZGWEZaRmZuL0ZpblEhRicvRlxvUSQwZW1GJy9GX29GW3BGYG9GYm8=for 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and compute the expected path length and variance for trees of size 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hnt: use the properties of probability generating functions to compute the expected value and variance). Finally, plot the the probability distribution for 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I.E. plot the points 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(for non-zero 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for 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using pointplot from the plots package. For each LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYn compute the expected path length and variance.