I would love to get an idea of how many deer we have on our property. Without hiring a surveyor or a helicopter, how does one do this on the cheap?

Is there a camera / land size formula?

Trailcams and spotlight counts. You can find information for the "how to" on spotlight counts on TPWDs website

TPWD should have information on what your area should support as in carrying capacity. Like fouzman said your going to have to actually survey your own property to get an accurate number, it varies depending on many things, drought, hunting pressure, predators etc.

depending on size of your place by the time you buy cameras, do the counts make sure everything works like it is supposed to bla bla bla....... its a wash to just pay for the helo survey.... usually ranches under 2K acres can be done in 2 1/2 hours or less, so about 1300ish depending on the bird you want to fly in. If you are going to do your own count on camera and/or spotlight, you can do your own count from the air. Most pilots are very helpful counting and collecting raw data... what you do with the data is another story.

Make sure you are not gathering info just for infos sake. We did one and the landowner refused to believe the results, so basically the info was useless. Some things are nice to know, but if the landowner doesn't recognize the information and its' usefulness, it is all a wash.

I would check into getting a drone survey done. Or buy a good one and do it yourself. PM JohnRussell he recently made the plunge and I'm sure he would be a good information source for buying or finding resources for having the survey done.