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That depends on the lens mount. Tamron makes lenses for a variety of cameras. If it has a Nikon mount, it fits onto the N2020 just like a Nikon lens. If it has, say, a Canon mount, then it won't fit at all.
Tamron makes their lenses with a variety of mounts. A lens with a Canon mount, for example, will not fit a Nikon. Assuming the lens has a Nikon mount then yes, it is compatible with the D60. However, the D60 lacks an autofocus motor, relying on the lens to have one. Some Tamron lenses have such a motor, some do not. Such lenses will not autofocus with the D60, but everything else still works, including manual focus.
i used d3100 a few months ago, for your information this camera only support autofocus with built-in motor lenses if you have interest on wildlife, you should use tele-lens whose focal length above 200mm if Nikon lenses cost you too much, you can choose other lenses from Tamron, Tokina, or Sigma
for family photograph, the kit lens is able to do this (18 - 55mm), for natural beauty, i myself use nikkor 50mm/f 1,8 which has great power in low light condition
Tamron is a third party lens manufacturer and make lenses to fit all popular makes of cameras both film and digital. I'm sure the top two would be Canon and Nikon. So if the lens you have there is a Nikon Mount it will fit Nikon Auto Focus cameras both film and digital
The Nikon D3000 does not have a focusing motor built-in, relying on the lens to have a motor. The Tamron AF 55-200mm does not have the necessary motor; it only auto-focuses with Nikon cameras that have the motor in the camera body.
For Nikon lenses, you need lenses designated AF-S. For Tamron, you need lenses designated USD and/or BIM.
That probably wasn't the answer you wanted to see, but there it is.