I am trying to do some proofs on my own in Graph theory. I just want someone to check if this proof is mathematically sufficient and correct in the first place. I have been struggling to determine whether my proofs are sufficiennt and mathematically correct or not. Do you have any tips on that? What is the boundary between something being sufficient and not?
If $T$ is a tree whose every vertex adjacent to leaf has degree of at least 3, prove that there exist two leaves who are adjacent to same vertex.
Find the longest path in a tree $T$. Endpoints of this path are both leaves. If they weren't leaves then we wouldn't have the longest path. That means that their neighbours have degree of at least 3. Let $u$ and $w$ be those endpoints. If neighbour of $u$ is a leaf then we're done, we have that two leaves who are adjacent to same vertex. If not, that means that we could build a longer path than a path between $u$ and $w$ which is a contradiction with our hypothesis that $uv$ is the longest path. Thus, if every vertex adjacent to leaf has degree of at least three than there exists two leaves who share the same parent / who are adjacent to same vertex.