True or false: let $T$ be a binary tree with $n$ vertices, then the sum of minimum child size can be upper bounded by $$\sum_{v \in T} \min(\text{size}(\text{left}(v)),\, \text{size}(\text{right}(v))) \leq O(n \log n)$$
The right-hand side is achieved by the balanced binary tree.
Note that $\sum_{v \in T} \text{size}(v)$ can be $\Omega(n^2)$ if the tree is a path, but this edge case doesn't apply to the above.