When transitioning my daughter from her high chair to a regular chair, we bought the Safety 1st Easy Care Booster Seat. It is a small booster seat that straps to a regular chair from both beneath and behind. We have had it on two different types of chairs in the house, and it has easily fit each one.
The Easy Care Booster seat comes with a small plastic tray that is detachable. We have never used this tray and it is easy to remove. I have never had a problem with the straps coming undone from the chair, and the safety straps that buckle the child in are secure.
The complaint I have regarding this booster seat is that the back is not a single piece, and comes off easily when my daughter leans back on it. I suspect it is removeable for use with older children, but because it is not attached well it is a hassle to deal with.