Let's define with respect to a binary tree $T$ (I'm sure that some notions are known, but I don't know the established names):
The "total height" $h(T)$ is the sum of all leaf heights of $T$.
An "full" $T$ has only even numbers of children for any node ($0$ or $2$).
A (left) "comb" is a special $T$ where all left children are leaves (and has the minimum number of nodes for a full $T$).
Example: The Wiki binary tree has $m=9$, is not full (exactly the nodes containing a $9$ are odd) and is obviously no comb. $h(T)=3+4+4+4=15$. It is not optimal: A comb would have, with maximum height $H=(m+1)/2$, $h=(H+1)(H+2)/2-2=19$.
Let $T$ be a binary tree with $m$ nodes. Which shape of $T$ gives a maximum $h$? I suspect it's a comb, which is also full. Assuming I'm wrong, which full $T$ gives a maximum $h$?