What are favorite foods for deer in nature and what might attract deer to graze on our propertyÉ

Apples...

Deer eat just about every kind of plant that they can digest, anything with tender leaves. Most native deer don't eat grass because they can't digest grass (They aren't ruminants like cows.) But they will eat everything from clover and rose bushes to poison oak. Sometimes they'll avoid certain things that are toxic or have a strong flavor, like rosemary or boxwood. But if they are hungry enough, they'll get less fussy and start eating things they don't normall eat... I live in California and they eat virtually everything I've planted outside my house (except for rosemary and boxwood!). I also spent two years studying what deer eat for a lab at UC Berkeley. I found almost 200 different species of plants in their scat... so take my word for it!
~Asking

I think oats and corn are one of the cheapest things you can buy in bulk to attract deer. Mind you if the deer come to your yard they may eat your garden, leaves of low bushes and trees, and other foliage, too.