The Local Authority will have an Assets Team who will be able to tell you definitively whether the tree in question is Council maintained or not, although if it's not council they more than likely won't be able to say who does. Data protection, Crown copyright, etc.
How you can actually get hold of someone in that Team and how much they will charge are different matters entirely! You can always start with having a look at the Land Reg site, but be mindful that the boundaries shown are a guide only.
Most Planning Authorities are now catching up with the modern age and have an interactive online map to show their trees subject to a Preservation Order and where their Conservation Areas are.