Most trees won’t grow that fast (well almost nothing grows as fast as bamboo). Usually height of a plant or tree is determined by the available shade (more shade, the more the tree will try to climb above it), and available water. Less water generally means shorter, and more (easy) sun usually means shorter, though there is a huge genetic element of course. Some trees will collect significant water from rain or fog when it’s hard to get water up the tree from their normal transport methods (like redwoods) because of their height.
A normal tree (which is slow growing) typically will get taller and wider each year until it starts maxing out something like water transport, or dies due to disease and competition from other trees.
The rings you are referring to are generally gathered from the base of the tree as that is the part most likely to grow (for sure). If you go to the top of the tree, it’s a bit harder to tell what has been going on.
And the ring itself is formed when the tree stops growing as part of the yearly seasonal cycle.
Bamboo doesn’t have rings.