Randomized Meldable Heap - Expected Height
In the paper, $h_Q$ isn't the height.  It's the length of a random walk away from the root in a full binary tree (they insist every leaf is "nil"), so the expression they have is the right thing.

Also, you can avoid induction.  The probability of ending at a specific leaf of depth $d$ is just $2^{-d}$.  So the expected length of the walk is 

$$
\sum_{\ell\in \text{leaves}(Q)} \text{depth}(\ell)2^{-\text{depth}(\ell)}
$$

which the entropy of a distribution a set of size $|\text{leaves}(Q)|$.