You link them to a schedule. They'll grow every 10 days you spend studying that schedule. Some trees take 40 days (5 stages), some 50 (6 stages). For example, a さくら at 32 days would be at stage 4/6.
To place the tree in the garden: My Stuff --> click on the tree --> arrange where you want it.
To make the tree grow: while still in the editing mode, click on the tree. In the popup there is a little sapling icon, which brings up another window. Use the drop-down to pick a schedule to link the tree to. Every day you study on that schedule, it counts toward growing your tree. Seems like ten days for every stage.