I have come across the following stochastic process which seems very elementary, although I do not know any appropriate terminology for it; I greatly appreciate any suggestions!

Suppose I am given integers $n$ and $k$, with $k\ll n$.  I create a binary tree as follows:

 1. Create a root node, with label $n$.
 2. While there exists a leaf $u$ whose label $\ell$ is greater than $k$, add two descendants to $u$ whose labels are $X$ and $X-\ell$, where $X$ is a binomial random variable having parameters $1/2$ and $\ell$ respectively.

I am interested in the basic distributional properties of this tree where $n\to\infty$ and $k$ is fixed, such as the depth and the labels of the leaves at termination.  Does this have a name?  I created one output below for $n=100$ and $k=7$:

[![Sequential binomial sampling with n = 100, k = 7][1]][1]


  [1]: https://i.sstatic.net/k9GWS.png