I am looking for a short proof of this fact. This is clearly true by drawing these trees, but I am having trouble putting it into writing. Somehow I need to select 3 of the 5 vertices and show that there must be a $3$-cycle or an independent set of size $3$. (Is this a Ramsey number or something like that?)
Is there an obvious/short argument here, or perhaps a lemma I could use to knock it out quickly?