HI Hawk:
First off, across is diameter and around is circumference. These measurements are done at breast height and labeled DBH (diameter at breast height) by those in the industry.
Depending on your species and requirements of light, known as shade tolerance, for optimum growth your spacing may vary somewhat. Also, insects and disease incidence and risk in your area will help you target the leave trees as well as marketability. Maybe some more risk can be taken with some species than others if they are higher valued species. Even that has it's pitfalls because market demands shift over short periods of time, so it's best to manage for the ecology of the area in my opinion.
At this stage in the game, since your diameters are small on that one sight (2 or 3 inches), you may consider doing a pre-commercial thinning with a brush saw. Down in your neck of the woods, I suspect it would be best done in the coolness of the winter months. Too hot in summer I suspect. If your doing at least 5 acres of ground, it would pay to get your own brush saw. It doesn't have to be a pro model, mid sized would be my recommendation on your acreage. To do a proper job, you should consult with your local agencies on criteria to follow for your species. The criteria would involve what I mentioned above, but also includes height, selection of quality crop trees, spacing. Remember to look up once in awhile so your leaving a tree with good form and vigour. Spacing might be a challenge to get a handle on, because if too many are cut, you can't stand them back up. So, what's most often done is have a goal of your target spacing, say 7 x 7 feet for instance, or target density is say 900 trees/acre. Make yourself a 6 foot pole or a broom handle and cut a length of rope to cover an area 1/100 of an acre (11' 9-1/4" for the rope length, square it, mult. by Pi). An easy figure so the density calculation can be done in a flash in your head. Swing the rope in a circle about that broom handle, count the trees (probably around 9 trees), multiply by 100, gives trees per acre. Another way to look at it is, each tree represents 49 ft2 (7'x7') of growing space. How many trees can fit an acre with that growing space? Well, 43560 ft2/acre divided by 49 ft2/tree = ~ 889 trees
9 trees x 100/acre = 900 trees/acre etc...
If you do a number of plots you could get a range of plot densities:
7,8,10,9,9,8,6,10,9,7,9,9,6,10 add up and divide by plots taken: 117 trees divided by 14 plots is 8.357. Mult. by 100 = 836 trees/acre averaged. Might want to tighten up a bit on the spacing, leave a few extra closer around holes and voids, but not closer then 3 feet.
Just a for example exercise.