Really, the concept of an expression or a parse tree sounds more complex than it really is.
It is just a piece of syntax turned into an object.
E.g. you can have something like a parenthesised expression, which consists of an opening parenthesis, an inner expression and a closing parenthesis.
That is literally an object with these three members representing this syntactic structure in whatever you parsed.
You can do that for all pieces of syntax in your language. Each piece can be turned into a class by simply making each part of the definition a property.
Then, the job of the parser is to turn your input into one single expression by piecing together all necessary sub expressions and assembling them.