Given a uniform recursive tree (URT) of size $N$ rooted at one node whereby the tree is generated as follows:
Starting with a root node, at each iteration, a new node is connected to one of the existing nodes with uniform probability until the total number of nodes is equal to $N$.
My question: (1) how can I find the probability of a node at each level is a leaf node (a childless node)? (2) how to compute the number of leaf nodes at each level given the size of the tree?
In the literature, there are some work on deriving the average hopcount or flooding time of such trees but I didn't find any specifically on finding leaf nodes. Pointers also would be very welcomed.
