text: abstract. following the model introduced by aguech, lasmar and mahmoud [probab. engrg. inform. sci. 21 (2007) 133–141], the weighted depth
  of a node in a labelled rooted tree is the sum of all labels on the path connecting the node to the root. we analyze weighted depths of nodes with given
  labels, the last inserted node, nodes ordered as visited by the depth first search
  process, the weighted path length and the weighted wiener index in a random
  binary search tree. we establish three regimes of nodes depending on whether
  the second order behaviour of their weighted depths follows from fluctuations
  of the keys on the path, the depth of the nodes, or both. finally, we investigate
  a random distribution function on the unit interval arising as scaling limit for
  weighted depths of nodes with at most one child.