Let T be a tree with 12 nodes. We know that it has exactly 3 vertices of degree 3 and exactly 1 vertex of degree 2. We do not know the degrees of the remaining vertices (besides the fact that none of them has degree 3 or 2).
(a) What is the sum of the degrees of all vertices in T? Justify your answer.
(b) Determine the degree sequence of T.
(c) Prove that the degree sequence given in item (b) is the only possible.
(d) Give two non-isomorphic trees with this degree sequence.
Regarding (a), I managed to solve it by the the property |V|=|E|+1 and the handshaking lemma. As a result the sum of the degrees of all vertices in T is 22.
However, I am stuck at (b), (c) and (d). Any help or hints would be very appreciated.
There are three vertices of degree $3$ and one of degree $2$, so we have "used up" 11 out of a total degree sum of $22$. Now there are eight vertices left, so we conclude that there must be one of these vertices with degree $4$ and the rest have degree $1$ (this is the only way to get eight positive integers to add to $11$ without using $2$ or $3$). I'll leave it to you to draw two nonisomorphic trees with set of degrees $\{4,3,3,3,2,1,1,1,1,1,1,1\}$.