Branches in Bucket Recursive Trees with Variable Capacities of Buckets

Trees are defined as connected graphs without cycles, and their properties are basics of graph theory. A tree on n nodes labeled 1, 2,..., n is a recursive tree if the node labeled 1 is distinguished as the root, and for each 2 ≤ k ≤ n, the labels of the nodes in the unique path from the root to the node labeled k form an increasing sequence [6]. Bucket trees are a generalization of the ordinary trees where buckets (or nodes) can hold up to b ≥ 1 labels. Mahmoud and Smythe [5] introduced bucket recursive trees as a generalization of ordinary recursive trees. In this model the capacity of buckets is fixed. They applied a probabilistic analysis for studying the height and depth of the largest label in these trees and Kuba and Panholzer [4] analyzed these trees as a special instance of bucket increasing trees which is a family of some combinatorial objects. Kazemi [2] introduced a new version of bucket recursive trees where the nodes are buckets with variable capacities labelled with integers 1, 2, ..., n. In fact, the capacity of buckets is a random variable in these models. He studied the depth quantity and the first Zagreb index in these models [3]. A bucket recursive tree with variable capacities of buckets (BRT-VCB) starts with the root labelled by 1 that has r ≥ 0 descendants each of them making a subtree. The nodes in the subtrees have capacities c < b or c = b. The nodes with capacities c < b are connected together with 1 edge and the nodes with capacities b have descendants ≥ 0 again each of them making a subtree such that the labels within these nodes are arranged in increasing order. The tree is completed when the label n is inserted in the tree. Figure 1 illustrates such a tree of size 19 with b = 3. For constructing a tree of size n+1 (attracting label n+1 to a tree of size n), if a leaf v has the capacity c < b, then we add the label n+1 to this node and make a node with capacity c+1 or produce a node n+1. But for a node with capacity b, we only produce a new node n + 1. The last nodes with c ≤ b labels at the end of subtrees are called leaves and other nodes are called non-leaves. The probability p, which gives the probability that label n+ 1 is attracted by node v in the model is c(v) n−|γ| , where γ = {v ∈ T ; c = c(v) = k < b, and v is a non-leaf}.