A binomial tree has two branches each with a probably of 0.5. Actually, p=0.5, and q=1-0.5=0.5. This generates a normal distribution with an evenly distributed probability mass.
Actually, we have to assume that each tier in the tree is complete. When we break data up into bins, we get a real number from the division, but we round up. Well, that's a tier that is incomplete, so we don't end up with a histogram approximating the normal.
Change the branching probabilities to p=0.9999 and q=0.0001 and that gets us a skewed normal. The probability mass shifted. That accounts for skewness.
Having incomplete tiers or bins less than 2^n generate binomial trees with areas that have no probability mass. This gives us kurtosis.
Response to comment:
When I was talking about determining the number of bins, round up to the next integer.
Quincunx machines drop balls that come to eventually approximate the normal distribution via the binomial. Several assumptions are made by such a machine: 1) the number of bins is finite, 2) the underlying tree is binary, and 3) the probabilities are fixed. The Quincunx machine at the Museum of Mathematics in New York, lets the user dynamically change the probabilities. The probabilities can change at any time, even before the current layer is finished. Hence this idea about the bins not being filled.
Unlike what I said in my original answer when you have a void in the tree, the distribution demonstrates kurtosis.
I'm looking at this from the perspective of generative systems. I use a triangle to summarize decision trees. When a novel decision is made, more bins are added at the base of the triangle, and in terms of the distribution, in the tails. Trimming subtrees from the tree would leave voids in the distribution's probability mass.
I only replied to give you an intuitive sense. Labels? I've used Excel and played with the probabilities in the binomial and generated the expected skews. I have not done so with kurtosis, it doesn't help that we are forced to think about probability mass as being static while using language suggesting movement. The underlying data or balls cause the kurtosis. Then, we analyze it variously and attribute it to shape descriptive terms like center, shoulder, and tail. The only things we have to work with are the bins. Bins live dynamic lives even if the data can't.