Let $T^n$ be a path with $2n$ edges rooted at its middle vertex.  If I understand your question correctly, then $S$ has **Property $m$** if and only if it does not contain $T^{m+1}$ as a rooted subtree.  Property $m$ can also be described globally, rather than forbidding local structure.  For example, $S$ has Property 1, if and only if it is a [Caterpillar_tree][1], where one end of the central path is the root $v_0$. Property $m$ can also be defined via generalizations of Caterpillar trees.  Namely, say that a tree is an *$m$-caterpillar* if all vertices are within distance $m$ of a central path.  Then $S$ has property $m$ if and only if $S$ is an $m$-caterpillar, where one end of the central path is the root $v_0$.  


  [1]: https://en.wikipedia.org/wiki/Caterpillar_tree