Let $T$ be a rooted tree. For any subtree $T' \subset T$ write $L(T')$ for the number of leaves of $T'$.

Further, for $T' \subset T$ define the *branch-depth* of a node $v \in T'$ as the number of nodes $w$ on the path from $v$ to $root(T')$ having more than a single child. The branch depth of $T'$ is then the maximal branch-depth of its leaves.

Let's call a tree binary if each node has at most two children.

I wonder if something along the following lines is true. There is constant $c > 0$ such that for any tree $T$ with $N$ leaves there is a subtree $T$ with $L(T) \geq N^{c}$ which is either binary or has branch-depth $O(\log N)$.