Expected cuts to strip a rooted tree

You have a rooted tree. At each step, pick uniformly at random one of the nodes still present other than the root, and delete it along with its entire subtree. Repeat until only the root is left. What is the expected number of steps?

$\sum_{v\ne\text{root}}\dfrac{1}{d(v)}$, where $d(v)$ is the depth of $v$ (root at depth 0). Node $v$ costs a step only if it is picked itself, which happens exactly when it is chosen before any of its $d(v)-1$ non-root ancestors. By symmetry among those $d(v)$ nodes this has probability $\tfrac1{d(v)}$; sum by linearity.