Assume I generate a random binary tree with a bounded height with $n$ nodes. For a given key we measure the length of its path (the maximum can be $n-1$). So my Question is what is the distribution of the path length?
The tree is constructed by recursively partitioning the given n points until all points are isolated or the tree height is reached. in each step of the recursion we choose randomly a split that is between max and min of the values that reach this node.
Would be great to know as much as possible regarding this distribution like expected value, variance and of course type and shape of the distribution.
