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 labelledrooted tree is the sum of all labels on the path connecting the node to theroot. We analyze weighted depths of nodes with given labels, the last insertednode, nodes ordered as visited by the depth first search process, the weightedpath length and the weighted Wiener index in a random binary search tree. Weestablish three regimes of nodes depending on whether the second orderbehaviour of their weighted depths follows from fluctuations of the keys on thepath, the depth of the nodes, or both. Finally, we investigate a randomdistribution function on the unit interval arising as scaling limit forweighted depths of nodes with at most one child.
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