A Walk Through Combinatorics: An introduction to Enumeration and Graph Theory by Miklós Bóna

Author:Miklós Bóna
Language: eng
Format: epub
Publisher: World Scientific Publishing Co. Pte. Ltd.

and comparing this to the result of part a, we see that N(Fk)= nk–1.

(c) Choose k= n, then Fk is the empty forest (n isolated vertices), and all rooted trees contain Fk. Then (10.1) shows that N(Fk)=nn–1, so this is the number of all rooted trees on [n.] The number of unrooted trees on [n] is therefore nn – 2.

(13) Keeping the notation of the previous exercise, N*(Fn)= nn–1 as a special case of (10.1). Now let N**(Fk) be the number of those refining sequences F1, F2, o o., Fn whose kth term is Fk There are N*(Fk) choices for the part F, F2,…, Fk of such a sequence, thenthere are (n –k)! different orders to remove the remaining n– k edges.

This shows that

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