None whatsoever. Leave gymnastics in the gym.
For an older and more advanced gymnast (like, at least 12 years old and at least level 7), MAYBE a small panel mat and a bare stretch of wall with nothing hard or breakable around it, for handstand work and shaping drills, and MAYBE a pull-up bar (but only if you can set it up in a way that ensures there's not enough space to swing on it). But even those are not necessary.
PLEASE PLEASE PLEASE FOR THE LOVE OF GOD, BUDDHA, CTHULHU, SIMONE BILES, THE FLYING SPAGHETTI MONSTER, OR WHOMEVER ELSE YOU MAY WORSHIP, DO NOT GET A BACKYARD TRAMPOLINE NOW OR AT ANY TIME IN THE FUTURE.
As a 20-year coach whose strongest focus is trampoline and tumbling, if I could outlaw backyard trampolines I would do so with zero hesitation. They are liability nightmares, they are technique nightmares, they are safety nightmares. They are right up there with nuclear weapons and automated phone trees as one of the worst things ever invented.