Do trees grow from the bottom or the top, or is that oversimplifying?
It appears you are mostly interested in the vertical growth of trees. As @Joe Healy points out, this is done only by cells in the apical meristem. This region always remains at the top tip of the tree, that is where the upward growth occurs. The apical meristem, as it continues upward with growth, will leave portions of itself behind that may result in branches (axillary merristems). So yes, unless you carve at the very tip of the tree, your initials will remain at the same height you carved them. 

A very simple experiment that I've done many times with students demonstrates this. Germinate some corn seeds until the shoot and the root are about 2 cm long. With a fine Sharpie, mark each with lines every 4 mm or so. Continue the germination. After a few more days you'll note that the line spacings nearest the kernal are the same as when you drew them. But on the very outboard end, near the tips of shoot and root, the spacing will appear stretched out.

Another type of meristem, the vascular cambium, is responsible for the increase in diameter of a tree. It is a sheath that lies between the bark and wood producing new wood to the inside and new bark to the outside.