∅-free regular expressions?
Preforming the proof by induction on the structure of the tree is good way to go about this. I like to think about how I would write a program to remove them because this directly leads to a constructive proof.

Say our regular expressions have the following form

$$
\begin{align}
regex ::=&amp; \,\, regex \,\, regex \\
        |&amp; \,\, regex^* \\
        |&amp; \,\, regex^+ \\
        |&amp; \,\, regex\,? \\
        |&amp; \,\, regex + regex \\
        |&amp; \,\, char \\
        |&amp; \,\, \varnothing
\end{align}
$$

The bases cases are the cases in which the regex is either a character or $\varnothing$

To give an inductive step say we have a regular expression of the form $regex + regex$. We can then recursively convert the two sub parts to regular expressions which either are the empty set or contain no empty set. So we have 4 cases

case 1: Both expressions contain no $\varnothing$ in which case we are done

case 2/3/4: One of the sides (possibly both) is $\varnothing$ so we remove that side. One needs to prove that this is valid but it isn't hard.

all the cases are going to precede as such with different proofs that the empty set can be eliminated.