Finding number of nodes of degree 1 in a tree Question: Given a tree with 15 nodes, no nodes of degree 2, the maximum degree is 5, and there is at least 1 node with degree 5 in the tree, show that the tree has either 10 or 11 nodes of degree 3.

A noder of degree $1$ in a tree is a leaf. If a tree has $n$ nodes and $m$ of these nodes $v_1,...,v_m$ have degree $\geq 3$ and all other vertices of the tree have degree $\leq 2$, then $2 - 2\cdot m + \sum_{\ell = 1}^m d(v_\ell)$ is the number of leaves of the tree. 
Does this help you to estimate your exercise?
P.S. The claim above can be proved by induction on $n$.
