What is the expected value of the expected distance of a leaf from the root of a full binary tree?

Question

Let $T$ be a full binary tree that is randomly chosen with the same probability from the set of all the full binary trees with $n$ leafs. What is the expected value of the expected distance of a leaf from the root of $T$, respective to $n$? For a fixed tree $T$, the expected distance of a leaf from the root can be deterministically calculated; that is $\mathbb{E}[\text{dist}(x, \text{root}): x \in V(T)]$; now that it's randomly chosen, we want the value of $\mathbb{E}[\mathbb{E}[\text{dist}(x, \text{root}): x \in V(T)]: T \in \text{BinaryTreesWithNLeafs}]$.