I just want to make sure i had this question right:
Count the number of trees on the set of vertices $V=\{1,...,12\}$ where $d(1)=d(2)=5,d(3)=3,d(4)=d(5)=...=d(12)=1$
*$d(1)=$ degree of $v_1$
What I did was to draw a graph according to these requirements and then write down the matching Prüfer sequence, which was (in my case alone of course) 3312222111. And so I was looking for the number of sequences of length 10 that has 4 2's, 4 1's and 2 3's and I got $\binom{10}{4} \cdot \binom{6}{4} \cdot \binom{2}{2}=3150$
Two questions:
- Did I get it right?
- Is there an easier way?
Thanks!