Below is what I sent to the manager of the arboretum. She did reply and said she'd pass along the information.
In the Texas Penal Code, 30.06 (the very section that prescribes
the method of the posting of these signs) it says under
subsection (e):
"(e) It is an exception to the application of this section
that the property on which the license holder carries a handgun
is owned or leased by a governmental entity and is not a premises
or other place on which the license holder is prohibited from
carrying the handgun under Section 46.03 or 46.035."
In other words, if it's government property and carry is not
prohibited for another reason (such as being a bar or a
courthouse, etc) then the 30.06 sign is unenforceable.
This does not mean the sign at the arboretum is illegal or
otherwise unlawful. But it does mean that it is completely
un-necessary and unenforceable.