I interpret the OP's question to be:
Taking it as given that Wattle is capitalised in this context, should tree in the phrase Wattle tree be capitalised as well?
That is, I set aside the question, raised in another answer, about the capitalisation of Wattle, and simply assume that, in this context, it should be capitalised.
The question is similar to the question that was asked before about the capitalisation of, for example, integral in Riemann integral. In all these cases we have a phrase that contains a word that is definitely a proper name, and a word that, in other contexts, functions as a common noun. The question is then whether to treat the latter in such cases as a common noun or a part of a two-word proper name.
There is no general answer to such questions; it is a matter of conventions that differ from one field to another. Consider, for example the geographic names, such as the Atlantic Ocean. In these cases, it is a well established convention to treat the whole two-word phrase as the proper name of the ocean, and to consequently capitalise both words. Somebody could reason that this is simply an ocean, one among several, that is distinguished from the others by the name Atlantic, and that ocean should therefore not be capitalised. There would be nothing wrong with this reasoning considered in the abstract (after all, that's precisely what is done in, say, French), it's just that the universally accepted convention among English speakers is to the contrary. The conventions, however, go the other way with respect to Riemann integral, Euler's number and suchlike. One could argue that Euler's number is the proper name of that number and that consequently both words in that phrase ought to be capitalised; there would be nothing in principle wrong about that argument; it's just that, as a matter of convention, it hasn't been generally embraced.
So, the answer to the OP is: there is no general rule of the language that determines whether to capitalise tree in Wattle tree (again, taking it as given that Wattle is to be capitalised). It is a matter of convention in the relevant field. If it is an established practice among the respectable writers in the field to capitalise it, it would be wise to go along; if not, not.