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Suppose I have the degree sequence $(5,4,3,2,2,2,1,1,1,1,1,1,1,1)$. This is a tree, and a simple representation is:

enter image description here

(My intuition says that any degree sequence that corresponds to a tree can be laid out like this in a caterpillar. @David demonstrated that is true).

Question:
Is there a good approach to counting all possible non-isomorphic trees that have this degree sequence?

Easier question: (which I would still like to see a good approach for)
How many non-isomorphic caterpillar graphs have this degree sequence?

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    Re "My intuition says...": see [here](http://math.stackexchange.com/questions/2145075). – 2017-02-15
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    I see there is [a generic version of this question](http://mathoverflow.net/questions/37665/degree-sequences-and-graph-enumeration) on math***overflow*** – 2017-02-15

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It looks like there's not going to be a clever way to do this. Using geng 14 13:13 -c (the package geng comes with Nauty), we can generate representatives from each isomorphism class of 14-vertex trees. I wrote a GAP script to identify the ones with the degree sequence [ 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 ]. It turns out there's 150 of them, drawn below (with the vertices ascribed their degrees):

enter image description here

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    Impressive result! thanks for investigating and persisting to this outcome. – 2017-02-20
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    My favorites are graph 112 (last on row 28) and graph 145 (first on row 37) :-) – 2017-02-20