  Rebalancing schemes for dynamic binary search trees are numerous in the
literature, where the goal is to maintain trees of low height, either in the
worst-case or expected sense. In this paper we study randomized rebalancing
schemes for sequences of $n$ insertions into an initially empty binary search
tree, under the assumption that a tree only stores the elements and the tree
structure without any additional balance information. Seidel~(2009) presented a
top-down randomized insertion algorithm, where insertions take expected
$O\big(\lg^2 n\big)$ time, and the resulting trees have the same distribution
as inserting a uniform random permutation into a binary search tree without
rebalancing. Seidel states as an open problem if a similar result can be
achieved with bottom-up insertions. In this paper we fail to answer this
question.
  We consider two simple canonical randomized bottom-up insertion algorithms on
binary search trees, assuming that an insertion is given the position where to
insert the next element. The subsequent rebalancing is performed bottom-up in
expected $O(1)$ time, uses expected $O(1)$ random bits, performs at most two
rotations, and the rotations appear with geometrically decreasing probability
in the distance from the leaf. For some insertion sequences the expected depth
of each node is proved to be $O(\lg n)$. On the negative side, we prove for
both algorithms that there exist simple insertion sequences where the expected
depth is $\Omega(n)$, i.e., the studied rebalancing schemes are \emph{not}
competitive with (most) other rebalancing schemes in the literature.
