Cutting down very simple trees

Isolating a Leaf in Rooted Trees via Random Cuttings

“…Driven from the inspection that all these important increasing tree families satisfy the equation Tn+1 Tn = c 1 n + c 2 , with fixed constants c 1 , c 2 , for all n ≥ 1, we will consider such trees in more detail. Throughout this paper we will call increasing tree families satisfying this equation very simple increasing tree families, since it turns out from the characterization given below that the defining degree-weight generating functions ϕ(t) are the same as obtained in [16]. We will give now an exact answer to the question, which degree-weight generating functions are actually defining very simple increasing tree families.…”

“…It turned out that exactly those tree families with ϕ(t) given by Lemma 1 have this property and could be treated with the recursive approach. In [16] such tree families are called very simple tree families. For the random variable Z n studied in the present paper things are easier.…”

“…The transition probabilities. The required transition probabilities q n,k as defined in Subsection 2.3 were already computed in [16] by a generating functions approach, which is also here sketched. We can define the value q n,k equivalently as the probability that the number of descendants of a node (where the node itself is counted) that was chosen at random from one of the n − 1 non-root nodes in a random tree of size n is k. We require also the auxiliary valueq n,k , which denote the probability that the number of descendants of a randomly chosen node in a random tree of size n is k. Introducing the generating functions…”

“…An analogous approach, with transition probabilities q n,n−k , was used in [16] to study for simply generated tree families the random variable X n , i. e. the number of cuts to isolate the root of the tree. There one had to make a strong assumption on the tree family in order to justify this recursive approach: it was necessary that randomness is preserved by cutting off a random edge, which means that starting with a random tree of size n and removing a random edge, the remaining subtree of size k containing the root is actually a random tree of size k in this tree family.…”

“…bounds for the second moment E(X 2 n ). Limiting distribution results of X n for some classes of so called simply generated tree families, which contain Cayley trees as a special instance (see Subsection 2.1), are given in [16]. The problem for so called non-crossing trees was considered in [17].…”