I want to prove the following:
A simple graph with average degree at least $t$ contains all trees on $t$ vertices as a subgraph.
I tried proving this by induction. I first showed that if a graph $G$ has average degree at least $t$, then any subgraph $G'$ formed by removing any vertex $v$ from it has average degree at least $t - 1$. Using induction, given a tree $T$ on $t$ vertices, I am able to remove a leaf $l$ from $T$ and embed the remaining tree $T'$ in $G'$. However, this does not guarantee that the parent of $l$ in $G'$ is adjacent to $v$, which hints that I need a more constructive approach.
How can I modify my argument to obtain a solution?
Remark: The Erdős–Sós conjecture is very similar to the above statement:
A simple graph with average degree greater than $t-1$ contains all trees on $t + 1$ vertices as a subgraph.