Height of a "random" binary search tree

Question

See also this question, but I'd like to know the simpler case which in that question refers to CLRS 12.4 (looked it up - no formula found!). So, assume you have a list $L=\{1,\dots,n\}$ and an initially empty binary search tree $B$. You insert a random element $e$ of $L$ into $B$ and delete $e$ from $L$. Repeat until $L=\{\}$. Is the height of $B$ at the end really $O(\log n)$? (And the average number of children $2$?) Just for fun, I created a random $n=50$ BST and its height was $10$.