Depth of binary tree with few single children The node of a binary tree is called a single child if it has a parent but does not have a sibling. The root is by definition not considered a single child.

Let $T$ be a binary tree of size $n$, and let $k$ be the number of vertices in $T$ that are single children.
Is it true that if $\frac{k}{n}\leq \frac{1}{2}$
then the height of $T$ is $O(\log n)$?
You can construct a tree with no single children which has height $(n-1)/2$:
.