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hw12sol

# hw12sol - Math 55 Discrete Mathematics UC Berkeley Spring...

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Math 55: Discrete Mathematics UC Berkeley, Spring 2009 Solutions to Homework # 12 (due May 4) § 10.1, # 3: a) The root is a . b) The internal vertices are a,b,c,d,f,h,j,q,t c) The leaves are e,g,i,k,l,m,n,o,p,r,s,u . d) The children of j are q and r . e) The parent of h is c . f) The only sibling of o is f . g) The ancestors of m are a,b,f . h) The descendents of b are e,f,l,m,n . § 10.1, # 13* a) There are 3 non-isomorphic unrooted trees with five vertices: the star tree, the chain, and one other tree. b) The three unrooted trees above can be rooted in 2 + 3 + 4 = 9 non-isomorphic ways. § 10.1, # 18: By Theorem 4(ii), the answer is mi + 1 = 5 · 100 + 1 = 501. § 10.1, # 22: The model is a full 5-ary tree. There are 10 , 000 internal vertices, corresponding to the people who send out the letter. By Theorem 4(ii), the total number of vertices is mi + 1 = 50 , 001. Everyone expect the root receives the letter, so 50 , 000 people receive the letter. There are 40 , 001 = 50 , 001 - 10 , 000 leaves in the tree, and this is the number of people who receive the letter but do not send it out.

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hw12sol - Math 55 Discrete Mathematics UC Berkeley Spring...

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