The Parks Department has a Frequently answered question web page that says:
Q:
Who is responsible for maintaining the tree in front of my house?

A:
The Department of Parks & Recreation is responsible for maintaining all public trees located within the street right of way. Trees located between the street curb and the sidewalk are always within the right of way. Trees located adjacent to the sidewalk on the building side may be public trees depending on the width of the street right of way.

The link to the Parks Q and A page is below. Please report a specific address if you want them to check a specific city tree.

http://www.enjoyburlington.com/Trees/FAQ1.cfm