Our flower beds and slate patio have tree-like weeds and are utterly
gross. We're looking for something that we can put down to kill the
weeds for longer than four weeks, and hopefully prevent them from
returning.

We've been using Roundup for years. It does kill the weeds, but they
come back quickly.

Please, someone suggest something better than Roundup.

We're in the Rochester, NY area in the US.