Consider a rooted embedded tree of $n+1$ vertices. It is known that around the root for small $r$, volume of the ball of radius $r$ grows like $r^2$. Now suppose we are given that a certain subtree is contained in the tree (somewhere we dont know) whose volume is small (say of order $n^{\epsilon})$ for small $\epsilon$. Now select a vertex randomly not contained in the subtree. Can we have similar volume estimate? Is anything known about this?
It is known that the expected number of vertices around the root at a distance exactly $r$ is roughly $r$. Can we say anything in the above setting as well? Any reference will be helpful.
