Correct.
"Sticks and stones may break my bones," and they aren't too kind to recycling plant machinery either.
It makes sense that rocks shouldn't be put in your curbside recycling, but you should keep grass clippings, leaves, bush clippings and all yard waste out of the recycling bin as well.
You might consider composting if you have the space.
Your trash and recycling provider probably schedules brush and bulky pickups once or twice a year, and in most cases you can call and ask for one if you need a pickup that doesn't coincide with the company's schedule.
If you live inside the Tucson city limits, go to tucsonaz.gov/es/brush-and-bulky for more information. Outside of Tucson, check with your provider.