How do trees grow in Terraria?

Solution 1:

It takes a somewhat random amount of time for trees to grow once they are planted. Assuming that all the growth conditions are met (as detailed on the wiki page you posted), it can take anywhere between 30 seconds to 5 minutes, in my experience.

Once a tree has grown, it will not get any taller, and it will not grow any additional branches. If you cut part of the tree down (or cut branches off), it will not regrow those parts.

Solution 2:

To complement Mage Xy's answer, regarding your situation:

I planted a dozen or so in a spaced-out row and was quite surprised to see one of these suddenly sprout into a tree around three times the character height (~9 blocks). Other planted acorns haven't grown for several in-game days.

Other saplings (probably) will not grow due a violation of rules #2 and #3 from Wiki page you posted. Because if saplings are placed too close to each other and one of them grows up, it will take 3 tiles of horizontal space efectively preventing ajusted saplings from grow due space shortage.
So be sure to have at least 3 open tiles between them.