Graph Theory Fun Problem
Let $G$ be the host graph and $T$ the tree. Try "growing" $T$ in $G$. At each step, mark in $G$ the subtree $T'$ of $T$ that has been grown so far. Using the bounds on $\delta(G)$ and $\vert V(T)\vert$, argue that we can always expand the marked subgraph if it is not yet isomorphic to $T$. 

Oh, but you wanted the Probabilistic Method. Here we go: Toss a coin $k$ times, throw out the results, and then do the above (old joke).