Here's 2 ways to construct a forest. Please tell me if these are legal:

(2-3), (5-7), (11-13).

(2-3-5), (7,11), (13).

So what do these look like? I've attached a picture of what (2-3) represents.

If these graphs match your criteria, I count 1 degree 1 vertex hanging off the "2" in the (2-3) graph, and 2 degree 1 vertices hanging off the "3". Continuing, I count 35 degree 1 vertices total. I get the same count in the second layout I suggested.