Let's say that we have a binary, not necessarily complete, tree. This tree is the result of a "duplication process" of an initial node (root).
So initially we start with a single node (the root) that has these duplication probabilities:
P[0] = "probability of not duplicating/becoming a leaf"
P[1] = "probability for the current node of having only one child"
P[2] = "probability for the current node of having two children" = P[0] - P[1]
According to these probabilities we let the tree grow until a certain stopping condition is met.
My question is: assuming we want to collect every node at a certain level l (possibly not the deepest one) and count how many of those nodes are leaves. How many leaves should I expect to count?
My answer would be that, in general, I would expect a number of leaves equal (on average) to P[0]*N_l, where N_l is the total number of nodes at level l.
Am I correct?