I wanted to know about the bonquet thingy, too. But it seems like no one can answer it. Your tama will just look at it and then question mark will appeared. I found out that the king's present is mostly a bouquet.
I've figured out a trick also. If your item list is full, the king will not give you present but he gives you gotchi points instead. The gotchi points given can be high sometimes.
What the King gives you is totally random, I think.
The item I receive more often than others are pencils. I don't know, why, though, because I really don't need all those pencils.
I've also gotten shovels, plants, make-up, chocolate hearts (and other food/snacks), bouquets, and rings (I like how the ring is as big as the tamagotchi ).