I have a six year old to pick up the coin cell at the bottom of the tree and return it to the top.  If you don't have one yourself, you might consider a mechanical return mechanism.  We considered several approaches but did not pursue them to completion due to said six year old and laziness but are confident you could do it.

Each approach used a motor removed from a small camera.  The motor was attached to a salvaged gear mechanism from an old inkjet printer.  In each case the coin cell was able to move the cell the required height but the number of repetitions was limited and the mechanisms add complications that require further engineering.

Approach One - Winch

A spool was glued to one of the gears and heavy thread used to move the battery up and down.  A completed mechanism might look something like this:

Approach Two - Pulley

Similar to above but use a pulley so the motor can be offset with possible geometry advantages.

Approach Three - Belt

Approach Four - Mechanical Linkage

Something like this: