Here is the question:
Let $G$ be a tree with $n$ vertices, and no vertex in the tree has degree $2$. Find a function of $n$ that indicates the lower bound of the number of degree $1$ vertices in the tree.
with handshake lemma and Euler's formula, I get $n_1 =2 + \sum_{k =3}^{\infty} (k-2)n_k$ where $n_k$ is the number of degree $k$ vertices. However, this result cannot tell me the lower bound of $n_1$.
I guess the result is $n -\big\lfloor \frac{n}{2} \big\rfloor$, but not sure how to prove this.
Please give me some hint, thank you.