MATH10001 Mathematical Workshop. Graphs, Trees and Algorithms Part 2. Trees. From Trees to Prüfer Codes

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1 MATH10001 Mathematical Workshop Graphs, Trees and Algorithms Part 2 Trees Recall that a simple graph is one without loops or multiple edges. We are interested in a special type of simple graph: A tree is a simple connected graph with no cycles.(i.e. no paths (u,v 1,...,v n 1,u) of length 1 from a vertex u to itself). Any vertex of degree 1 in a tree is known as a leaf. Trees with 1,2,3 and 4 vertices In this lecture we will attempt to answer the following question. How many trees can we construct on n vertices, labelled 1,2,...,n? This question was originally answered by the English mathematician Arthur Cayley in We will prove Cayley s theorem, not in the same way as Cayley but using a method devised by the German mathematician Heinz Prüfer. His approach is to use an algorithm to construct a code from a tree. By showing that there are the same number of codes as trees we can then enumerate the trees on n vertices. From Trees to Prüfer Codes We now explain how to encode our trees as sequences of numbers. For simplicity we will assume our vertices are labelled {1,2,...,n}. A Prüfer code on n > 2 is a sequence of numbers (s 1,s 2,...,s n 2 ) where s i {1,...,n} for each i = 1,...,n 2. For example (3, 1, 1, 4, 4, 4) and (3, 4, 5, 6, 7, 8) are two Prüfer codes on n = 8. 1

2 Algorithm PC1 (Prüfer Code 1) INPUT: a tree T with vertex set {1,...,n} where n > 2 OUTPUT: a Prüfer code PC1(T ) on n. (step 1) i = 1 (step 2) j = smallest leaf of T, and suppose the incident edge is e = jk (step 3) s i = k and T := T with e and j removed. (step 4) if i = n 2, output (s 1,...,s n 2 ) = PC1(T ) and STOP; otherwise i := i + 1 and go to step 2. Example Run PC1 on the tree T with adjacency table: ,6 4 2, ,3,4 From Prüfer Codes to Trees We now introduce an algorithm that performs the reverse task to PC1. We need the following notation: for a Prüfer code (s 1,...,s n 2 ) on n let o( j) be the number of occurrences of the number j in the Prüfer code, for j = 1,...,n. Algorithm PC2 (Prüfer Code 2) INPUT: a Prüfer code P = (s 1,s 2,...,s n 2 ) on n OUTPUT: a tree with edge set E and vertex set {1,...,n} (step 1) i = 1 and E = /0 (step 2) j = the smallest integer with o( j) = 0 and E := E { js i }. (step 3) o( j) = 1 and o(s i ) = o(s i ) 1 (step 4) if i = n 2 go to step 5, otherwise i := i + 1 and go to step 2. (step 5) E := E {kl} where o(k) = o(l) = 0 and o( j) = 1 for all other j. (step 6) output PC2(P) = T the tree with edge set E and STOP. 2

3 Example Run PC2 on the Prüfer code P = (6,4,6,3) Theorem 1 For any Prüfer code P and tree T, P = PC1(T ) if and only if T = PC2(P). Proof By induction on the number of vertices. Details not included in this project. Theorem 2 There are n n 2 trees with vertex set {1,2,...,n}. Proof By Theorem 1 the number of trees with vertex set {1,2,...,n} is the same as the number of Prüfer codes on n, ie. n n 2. 3

4 MATH10001 Mathematical Workshop Part 2 Problems Problem 4 Draw diagrams for all trees, up to isomorphism, with 5 vertices. For each diagram, how many trees are there with vertex set {1,2,3,4,5}? Problem 5 Let T be a tree with n vertices. (i) Prove that there is a unique path between any two vertices in T. (ii) Use induction on n to prove that T has n 1 edges. (Hint: for the induction step, remove an edge from T and consider the resulting graph.) Problem 6 Run PC1 on the tree T with adjacency table: , ,6,8 6 3, ,5,7. Then run PC2 on the output to get back to T. Write out all the steps in both algorithms. Project Report The assessment for this project is by an individual project report. The report should contain your solutions to the six problems and the homework for part 1. Your report should be well presented and the problem solutions should be clearly explained. Even though you have worked as a group on this project, the report should be all your own work. There are marks for the clarity as well as the correctness of your mathematical arguments. Please hand in your report to Alan Turing reception by 1pm on Friday 14th December. You should attach a cover sheet to your report and put your group number on the front page. 4

5 There are 25 marks for this project. (a) 16 marks for the solutions to the problems. (b) 3 marks for the homework. (c) 2 marks for the presentation of your report (clear explanations and layout). (d) 1/4 of the average mark for your group for (a) out of 4. Any student who does not attend one group session, without good reason, will get half the marks for (d). If a student misses both group sessions, without good reason, they will get 0 marks for (d). Please notify the School as soon as possible if you miss a session and fill in a Self Certification Form available from the Alan Turing Building reception. 5