I have the following algorithm for "constructing" a binary tree:
- A probability $p_g$ for elongation, i.e. adding an edge
- A probability $p_b$ for branching, i.e. adding to a node two "child" edges
Also it holds that $p_g$ + $p_b$ = 1.
- A resource value R, which serves as a way for my binary tree algorithm to halt: For each elongation, the resource value is reduced by one. At each branching point, a uniform random number $x$ from the unit interval is drawn, and then multiplied by the resource, whereas one side of the left child-edge receives x*resource and the right child-edge receives (1-x)*resource.
This procedure is continued until the resource is used up.
Now what I am inerested at is the distribution of the branchlengths in my binary tree, whereas a branchlength is defined as the length between two branching points. I assume that my branchlength accords to the Random Variables of a geometric distribution. Is it further correct to assume, that the resource parameter does nothing other then to define the support of the geometric distribution I am looking at? I was wondering since I think that the resource can be understood as a halting probability, right? I.e. instead of resource, I could also introduce a halting probability $p_h$. Would then subtrees with different values of resource correspond to subtrees with different halting probabilities, and hence their branchlengths to the random variables of geometric distribution with different parameters?
Thanks