Sapling
A Senju gifts the person they are courting with a sapling. If the plant grows to maturity then they're compatible. If it dies then their love wasn't meant to be.
This ties into my personal headcanon of all Senju having a sort of Mokudon-lite. Their chakra infuses the plants they grow and those plants maintain a few special properties. One of them is that they passively absorb the chakra of the people around them.
For the sapling to thrive: The person attending to the plant has to have a decent amount of chakra, have chakra compatible with the giver, and also be near the plant often.
If the person doesn't have a good amount of chakra, then their children are at a disadvantage to learning the shinobi arts. If their chakra isn't compatible then there could be a problem with reproducing in the first place. If they aren't near the plant/taking care of it often, then they likely value the relationship less than the Senju who gifted it to them.
Of course, nobody really knows the exact reasons behind the superstition. There was just a history where a pattern of dead sapling == failed partnership showed up and they drew their own conclusions.
Tobirama gifts Izuna a plum tree upon the formation of the village. It's meant as a sign of peace and goodwill at first. He completely forgot about the tradition until much later. Coincidentally, when he'd realized he actually did want to start a relationship with Izuna...
But, obviously it wouldn't work out between them - better to just remain friends! At least, that's what he thinks, before stepping foot in the backyard of the Uchiha main house and seeing a beautiful flowering plum tree.
