Since 8 December 2015 Broomhill, including the greenspace, has been a conservation area.
Conservation Area status requires the character of an area to be maintained, but recognises the need for change, requiring a sensitive approach that respects, conserves and enhances the special character and attractiveness of the area.
With regard to trees the document says:
Demolition of any structure and works affecting trees are also subject to Planning Control. Trees are automatically protected and you must give the Council 6 weeks notice in writing or via Email: TreeEnquiry@drs.glasgow.gov.uk of the intended works.
Notably, the greenspace at Broomhill Avenue is identified on the Broomhill Conservation Townscape Character Assessment Plan as a Green Corridor (see p.18 at the link below).
You can view the document describing the Broomhill Conservation area here: https://www.glasgow.gov.uk/CHttpHandler.ashx?id=32417&p=0