Consider a rooted tree of height $h$, such that all the leaves are at last layer. We perform the following random process: each edge is deleted with probability $0.5$, and otherwise it is retained. We are interested in the probability that after the process ends, there remains a path from the root to one of the leaves. In particular, we are interested in whether the probability goes to zero as $h$ goes to infinity or not.
It is known that if the tree is a complete $d$-ary tree for $d > 2$ then this probability is greater than $0$ regardless of $h$. However, I am interested in the case where the graph is irregular, and in particular may have internal vertices with a large number of children and may have internal vertices with only one or two children.
Suppose that we know that the average degree in each layer is large: Say, there is some large constant $d$ such that in every layer, the average number of children of vertices in that layer is at least $d$. Suppose that we also know that the maximal degree of a vertex is bounded, i.e., the number of children that a vertex can have is at most some constant $D$ which is independent of $h$ (and may be significantly larger than $d$, e.g., $D= 2^d$). Can we prove that, the probability that there is a path from the root to a leaf does not go to $0$ when $h$ goes to infinity (assuming $d$ is sufficiently large)? If not, can we say something about the rate of convergence (e.g., the probability is at least $\frac{1}{\log h})$?.