Washing machine running water coming out of toilet

Sounds like you have a partial blockage downstream from the toilet, Are you on city sewer or a septic tank? Back to you. Tom

I am on city sewer. The plumber came out yesterday and could not find a blockage. Another plumber is coming today to see if there are tree roots in the pipes. What the solution to that problem?

Solving a tree root problem means locating and repairing the break the roots are entering. However there is a way to control them.
Roots seek the nearest source of water, in this case a break in your sewer line. That old terra cotta sewer should be replaced by a new PVC one or at the very least someone should run a snake out until it hits roots. The snake should then be marked and taken outside and laid on the ground over the old sewer line. That will tell you where to dig to fix the break in the line for a temperary repair until you can get at least three estimates to replace with PVC.
For immediate relief first auger out the roots. To kill the existing roots and prevent further growth you can purchase RootX or Robics Foaming Root Killer that contain Dichlobrnic.
A less expensive way would be to call around to garden supply stores and ask for fine grain Copper Sulphate. Put a 1/2 pound in your toilet and flush it down. Repeat in 6 months.
Let me say that sooner or later you're going to have to bite the bullet and replace that old sewer line with PVC.
Hope this helps and thank you for rating my reply.