Palm Tree would be alright. Tear them up if they are fan palm types while they are green and then let them dry. Or if they have not been trimmed and have dead stuff hanging down, just cut if off with hedge clippers. If they are the King/Queen Areca types that have long fronds with leaves along the edges, you could just tear them off... I don't know what kind of Jasmine Tree you have, our Jasmines out here are all vines or shrubs. But some, I believe are poisonous and not good for animals to eat. I don't think the Jasmine shrub with the white flowers is poisonous, but here we have more than one kind of plant that is called "Jasmine". Down side of Palm Trees is they don't shed the litter by themselves. Somebody is going to have to cut it off... And you have to be careful because some of them are very thorny. If you have any Cypress trees, like the Italian cypress, they shed a lot of litter and you can scoop it up beneath the tree, unless it is a very clean garden and someone rakes it up all the time...
