Unlike the NCAA Tourney, #1 doesn't necessarily play #16 (or #32)
So, tennis tournament draws follow some mathematical rules, but also incorporate a bit of random chance to try and keep things interesting and fair. It seems like you already understand how seeding works so I won't bother explaining that - but I would like to point out that certain tennis tournaments sometimes follow different rules for determining the seed for a player and don't just strictly follow the ranking = seed formula that other tournaments use. For example, Wimbledon has their own formula that incorporates the player's performance in matches played on grass since that's the surface Wimbledon is played on.
Let's first ask ourselves - what is the point of having a methodology for how a tournament draw is constructed? The answer is - the main purpose is to preserve match-ups between highly seeded players until the later rounds of the tournament. It's a way for the "difficulty of opponent" quotient to increase as the tournament progresses. Knowing that, we can understand why - for every tournament - the #1 seed is placed at the top of the bracket and the #2 seed is placed at the bottom of the bracket. This ensures that the only match those two players can meet in is the final and attempts to keep those top seeded players participating in the tournament for as many rounds as possible (and $ell a$ many ticket$ a$ po$$ible).
After the #1 and #2 seeds have been placed, things get a little more random, but not by much. The scale of the formula changes depending on the size of the bracket but for the purpose of this answer I'll look at last years (2016) Wimbledon Men's Singles Draw.
In this draw you can see that Novak Djokovic was seeded #1 and placed at the top of the draw in position 1 with Andy Murray seeded #2 and placed at the bottom in position 128. In keeping with the theme of preserving big name matches until later rounds - seeds #3 and #4 are put into their own "quarters" of the draw to ensure that they won't play the #1 or #2 seed until the semi-finals at the earliest. This is why #3 seed Roger Federer is placed in the second quarter of the draw in position 33 and #4 seed Stan Wawrinka is placed in the third quarter of the draw in position 96.
Keep in mind that after the #1 and #2 seeds are placed - the rest of the placing of seeds into the draw can be somewhat random and different. In the example I just gave, Federer and Wawrinka's positions (33 and 96) could have been swapped depending on the randomness of their names being chosen - by someone pulling their name from a hat (at all 4 majors anyway). Different tournaments use different methods for name selection and have changed those methods historically at certain times. Here's a link to a draw ceremony video of the 2015 Australian Open Women's Singles draw if you care to watch.
So, back to the placement of seeds. Keep in mind the idea of "quadrants" - so 4 blocks each having 32 places in a tournament bracket size of 128. After the #1 and #2 seeds are placed, the rest of the seeds (#3 through #32) are placed somewhat randomly, but in specific positions in each quadrant of the draw. Again, those positions help to ensure matches between the better players not happen until the later rounds. You can notice a pattern to those positions mathematically. They are 1, 8, 9, 16, 17, 24, 25, 32, 33, 40, 41....down to 104, 105, 112, 113, 120, 121 and finally 128 (for the #2 seed).
If every match went according to plan with the (higher) seeded player winning, this placement of seeds into the bracket would lead to:
- Seeds 25-32 meeting seeds 1-8 in the third round
- Seeds 9-16 meeting seeds 17-24 in the third round
- Seeds 1-8 should meet seeds 9-16 in the fourth round
- Seeds 1-4 should play seeds 5-8 in the quarter finals
- Seeds 1 and 2 would play either 3 or 4 in semi finals
- Seeds 1 and 2 would play in the final.
There are also some guidelines that certain tournaments follow with regards to positions reserved for Wildcard and Qualifier players but I'll leave that for another time since this answer is already pretty lengthy.
The NCAA tournament (and therefore the seedings) are only run once a year. There might be one or two strong teams near the top from one tournament to the next, but much of the bracket will change dramatically.
In tennis, a few strong players might be in very similar relative positions for many months, covering many tournaments. Imagine if you were right around #8 during a run where someone that you didn't match up well against was holding #1 for some time. In a straight-seeding, you might be playing them in the quarter-finals with some consistency. Your game might give you the chance to beat #4 or #3 occasionally, but you might rarely face them if everyone is attending and #1 takes you out.
By changing the bracket from straight seeding, the tournaments randomize things a bit to avoid excess repetition of any particular matchup other than #1-#2.