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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:

  1. Did I get it right?
  2. Is there an easier way?


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1 Answer

up vote 1 down vote accepted

The answer appears to be correct. I got it by a completely different analysis. Vertices $3$ through $12$ are leaves, so the interval vertices are $1,2$, and $3$, and they must form a tree after all the leaves have been removed. This can happen in just three ways: $1-2-3$, $1-3-2$, or $2-1-3$. (The other six permutations are simply the reversals of these.) If vertex $3$ is in the middle, it gets one leaf, and the remaining eight leaves are split equally between vertices $1$ and $2$. If vertex $1$ or $2$ is in the middle, vertex $3$ gets two leaves, the middle vertex gets three, and the remaining internal vertex gets the other four. Thus, there are


possible labelled trees satisfying the condition on the degrees.

Your approach may be more efficient in general; mine works well here because there are so few internal vertices.

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That's nice, thanks –  yotamoo Mar 12 '12 at 9:53
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