What is the most efficient wood based fuel?

Which form of wood is actually the most efficient in furnaces? I would appreciate if there is a descending list of efficiency.

This includes any thing that can be made from wood, including charcoal though if it requires another unit of fuel to make it, do subtract from its efficiency.

I am aware of the similar question that is the fuel ranking. However, this question is specifically asking for a least-input greatest-output analysis for wood-only materials, to minimize the resource cost of getting efficient furnace fuel.

Its also to resolve a common argument of sticks vs planks and sticks vs crafted alternative.

So as mentioned by Schism, this is a input vs output question.

I made a chart!

The cost is calculated by assuming one Wood costs 24 (to avoid bad decimals). Planks therefore cost 6 (since one Wood makes four Planks), and Sticks cost 3 (since we make four sticks from two planks, or half a Wood per four-stack, making an eighth of a Wood per individual stick).

Charcoal is the most efficient, but it's kind of special. We need to kick off charcoal by first burning something else. This is in the table as Charcoal*. (See discussion in A word on Sticks vs Planks.) After that, we just use 1/8th of a Charcoal to make the rest of our Charcoal. This follows the progression c_n+1 = 24 + c_n/8. As n approaches infinity, the cost approaches 192/7, which is in the table as Charcoal (∞th).

Item            Time  Cost     Value
Charcoal (∞th)    80  27.4...  7.000
Charcoal (1st)    80  30       6.400
Plank             15   6       6.000
Fence (Console)   15   9       4.000
Stairs            15   9       4.000
Stick              5   3       4.000
Fence (PC)        15  10       3.600
Pressure Plate    15  12       3.000
Shovel            10  12       2.000
Sword             10  15       1.600
Crafting Table    15  24       1.500
Fence Gate        15  24       1.500
Wood              15  24       1.500
Axe               10  24       1.000
Pickaxe           10  24       1.000
Trapdoor          15  36       1.000
Chest             15  48       0.750

Time is the time in seconds that an item will burn for. (It takes 10 seconds to smelt a single item.) Cost is as described above. Value is Time / Cost * 24 / 10, or the number of items you can expect to smelt using 1 Wood's worth of that item.

^{Note: Fences are crafted with 6 sticks for 2 fences on Console. On PC, we craft 4 planks and 2 sticks to make 3 fences. The Console recipe is slightly more efficient, causing a split in the table.}

A word on Sticks vs Planks

From the table, using Wooden Planks is clearly more efficient than using Sticks. However, this is not true if we're interested in bulk-smelting Charcoal.

In theory, it would be superior to use Planks -- after all, they are more efficient than Sticks. However, in practice, there's no point to crafting the other two Charcoal at reduced value. This means that in practice, the Plank we use for the first Charcoal is really only worth 10 seconds of smelting time, giving it the same efficiency as that of Sticks.

We can calculate this: using Planks to make 3 Charcoal gives us a value of 6.857(3) + 7(n - 3), where n is the amount of Charcoal we'll end up using. By making only one Charcoal, we get a cost of 30 for the first Charcoal, giving us 6.4 + 7(n - 1), whether we use Sticks or Planks. However, for any positive, integer value of n (which comprise all valid values), the second expression is greater than the first. It is always better to craft only one Charcoal. Therefore, when crafting Charcoal, it doesn't matter whether you use Sticks or Planks.

tl;dr Planks are better than Sticks if you're directly using them as fuel. However, if you're making Charcoal, then they're equally good.