On the node structure of binary search trees

The external nodes of a binary search tree are of two types: arm nodes whose parents have degree 2, and foot nodes whose parents have degree 1. We study the positioning of these two types on the tree. We prove that the conditional distribution of the insertion depth of a key given that it is inserted in a foot node equals that of the conditional distribution given that it is inserted in an arm node shifted by 1. We further prove that the normalized path length of the arm nodes converges almost surely to 1/3 times the limit distribution of the normalized path length of all external nodes.