Expected number of subtree removals

You are given a rooted tree with $n$ nodes. At each step you choose a node uniformly at random from the nodes that are still in the tree, and remove that node together with its entire subtree. This continues until every node has been removed (that is, until the root is chosen). Find the expected number of steps.

Compute it for the complete binary tree with $7$ nodes: a root, its $2$ children, and $2$ children of each of those, for $7$ nodes in total.