How deep in the soil can a seed be placed and still develop into a plant/tree?
The general mathematical relationship requires an allometric equation. There are not as many for seedlings as there are for mature trees (but see the answer by @Richard-smith for a specific example). But lets start with a basic assumption that the emerging seedling will be a very tall cone (tall = high h to r ratio).

The volume of a cone is:


  $V = \frac{1}{3} \pi r^2h$


Then we need to know the density of the seedling to figure out how much carbon a seedling of height h would require. Lets say that the density is 1.5 g/cc (excluding water). 

Then, assume that the biomass is 1/2 carbon (a fairly reasonable assumption).

To figure out how tall a seed could grow given carbon reserves, it would be necessary to determine the amount of carbon in storage in the seed. Say there were 5 grams. There will be a carbon cost of reallocating this to growth, say 50%. So, for a seed that has 5g of carbon, its total volume might be calculated as:

 (5g * 50% growth respiration)/(50% C content of biomass) * 1.5 g/cc = 7.5 cc


assuming that the seedling will have a radius of 0.2 cm, the height could be calculated as:


  $h = \frac{V}{(3 \pi * 0.2^2)} 
    = \frac{V}{0.38}$
          = 20 cm (for V = 7.5)


Note: this is just a very quick "back of the envelope" calculation with many assumptions. C is 50% of plant biomass is pretty close (the actually number is closer to 48%). The growth respiration cost is a very rough estimate, and will vary with nitrogen content of the tissue, soil temperature, and etc.. The plant will likely need carbon left to build a leaf, and another resource may be limiting.