I came across this problem and I'm really struggling with it. Let $G$ be a simple graph with minimum degree $k$. Then $G$ contains a copy of every rooted tree of $k+1$ vertices.
I lack good intuition in most graph problems, so I tried induction on $k$, on the number of vertices and on the number of edges. For example, when trying to apply induction on $k$ I don't know how to justify that I can delete edges until the graph has minimum degree $k-1$.
I can't find a general argument without doing complicated cases or being too informal.
Any help would be much appreciated!