Will villagers breed if I trade with them but don't give them crops/food?

Question

Part of a villager breeder I'm making requires a lot of carrots to move in a square formation around villagers, and from some tests I've done with designs, the villagers are going to go through the carrots quickly. I'm planning on making a big farm for carrots for them, but it's all going to be a big hassle.

The Minecraft wiki says:

Villagers do not have a food item; they will breed of their own accord (and in their own time) depending on the available houses, or at least "registered doors", (See the village page for full details.) and whether or not they have bread, carrots, potatoes, or beetroots in their inventory, or have been trading with the player.

It says whether they have the crops or have traded with the player, so does that mean I can trade with them once and not have to worry about giving them food?

Answer 1

If you want them to breed once, then yes.

"Willingness" is cleared when producing a baby, and villagers will then need to be made willing again. Villager becomes willing when trading an offer for the first time, or with a 20% chance when trading an already traded offer.

So no, if you want to build a real breeder, there's no way around mass-producing food. Check out this excellent video of a carrot farm by UnaryBit (and a much longer tutorial with fixes, although original one still works). I've just made one in 1.11.2, working greatly even inside village boundary.