You have two different issues in your question. 1) How much bigger than the root ball should a grow box be? 2) Encouraging horizontal root growth by attaching a wooden plate to the bottom of a tree (or at least that's what I think you mean).
Grow boxes encourage growth by providing good drainage and air flow to the roots (when the proper medium is used). I would say you need the box to be greater than 2" around the root mass for that to occur. Now if you attach a piece of wood or tile to the bottom of the tree it kind of messes up the drainage and air flow a bit. If you do that, I would say the box needs to be a minimum of 3 - 4 inches greater than the dimensions piece of wood or tile to get any benefit. Maybe others will offer more information as well.
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