Distance in bucket recursive trees with variable bucket capacities and given number of leaves

A size-n bucket recursive tree Tn with variable bucket capacities and maximal bucket size b starts with the root labelled by 1. The tree grows by progressive attraction of increasing integer labels: when inserting label j + 1 into an existing bucket recursive tree Tj, except the labels in the non-leaf nodes with capacity b all labels in the tree (containing label 1) compete to attract the label j + 1. For the root node and nodes with capacity b, we always produce a new node j + 1. But for a leaf with capacity c b, either the label j + 1 is attached to this leaf as a new bucket containing only the label j +1 or is added to that leaf and make a node with capacity c+1. This process ends with inserting the label n (i.e., the largest label) in the tree. In this paper, we derive the distributional decomposition of the random variable ∆n,j, which counts the distance between label j and label n in a size-n bucket recursive tree with variable bucket capacities. Also, we show an asymptotic normality of this quantity and give a convergence in probability related to this random variable