I've used rooting hormone in the past, only for annual plants, I never tried it on
trees. The drawback is preventing the cutting from wilting as it's source of
water (the parent plant or tree it was taken from) has been severed. I'm willing to bet a cutting from that mulberry tree dipped in rooting hormone and placed outside in the soil will wilt and die almost immediately. Having said this, have you looked into air layering? It's a process of taking a small end of a branch of a tree or shrub, your mulberry for example, and getting it to form new
roots before it is ever cut from the parent tree. I think this will be a much more successful way to get a clone of that mulberry tree established in your yard. Here's a
thread with a
video about it and some step by step instructions:
https://permies.com/t/84300/Good-method-air-layering#697447