This is a question that David and I have been asked many times in the past. At last a new scheme has been introduced that will enable you to do this! Here is advice on the scheme from the City Council.

As part of National Tree Week, Oxford City Council is offering residents, businesses, schools and community groups the opportunity to adopt a tree. Other trees have been planted in our parks in remembrance of loved ones.
There are three options available, from watering and looking after an existing tree outside your property to purchasing and watering new trees in one of our parks.
National Tree Week is held from Monday 7 to Sunday 13 March and is organised by The Tree Council. They were founded in 1974 and launched National Tree Week in 1975.

There are three options available in the adopt a tree project, from watering and looking after an existing tree outside your property to purchasing and watering new trees in one of our parks.
Our parks service will discuss potential locations and species of tree, purchase your tree – average height 6-10ft, plant and stake your tree and provide you with some basic information on how to look after your tree
Once planted we would like you to help look after the tree by, maintaining and caring for the tree, watering and weeding, report any damage or unsafe trees.
Information and application forms are available from Oxford City Council’s Parks Service on 01865 252240, email parks@oxford.gov.uk or visit the website