Prove: 

If a tree has exactly $k \geq 1 $ nodes with degree $4$, then this tree has at least $2k +2 $ leaves. ( nodes with degree $< 4 $ are only allowed for the leaves ).

So I think that we can solve this with induction.

$ k = 1$ :

So we have exactly one node with degree $4$. this node has to have at least $4$ leaves, because of the degree. ***Or am I wrong?***

$ k \rightarrow k+1$:

We have a tree with $k$ nodes with degree $4$. We know that we have at least $2k+2$ leaves. We replace one of the leaves with a node and build a node of degree $4$. So we "lost" a leave but "gain" 3 new leaves. So we conclude that we have a tree with $k+1$ nodes with degree $4$ and $2k+2-1+3 = 2k+4 =2(k+1) +2 $ leaves. **Am I right?**