Bonsai trees are created by going to your pet's garden and pulling a fully grown tree out of the ground. Doing this brings up three options to decide the fate of your tree. Select the "Make Bonsai" option, then choose they type of tree pot you would like. Bonsai pots may cost coins or Playfish Cash, there are no current ways of turning a tree into a bonsai for free.
Once you have created your bonsai, you can place the item either in your chest or use it as decoration around your pet's home.
Once turned into a bonsai, your tree should no longer produce fruit. When you turn your tree into a bonsai, if there was fruit still on your tree, it would appear on your bonsai version. However, some people don't like fruit and will pick their tree clean, then turn it into a bonsai. Unfortunately, there is a glitch where the bonsai will still grow fruit, but this fruit is not harvestable. People have ended up with fruited bonsais when they did not want them.
According to the Playfish website: "Please be aware if you own a Beanstalk that making it into a Bonsai will also remove your Secret Beanstalk Room, so Sirius recommends you avoid doing this!"
Bonsai Pot Types
Here are some photos of Bonsai trees. Feel free to add your own!