Hook length formuli and their invariance properties?
Consider two posets $P,Q$ and an element $x \in V(P)$. Let $Pe(x)$ denote the immediate predecessors of $x$ and let $Su(x)$ denote the immediate successors of $x$ in $P$. Let $Min_Q,Max_Q$ denote the minimal, resp. maximal, elements of $Q$. Fix two mappings $Conn_1 : Pe(x) \rightarrow Min_Q$ and $Conn_2 : Su(x) \rightarrow Max_Q$. It is then possible to define a poset $R$ by starting with $(P - x) \uplus Q$ and by adding an arc $(w,Conn_1(w))$ for each $w \in Pe(x)$, and an arc $(w,Conn_2(w))$ for each $w \in Su(x)$.

This operation has a simple meaning for trees: given a tree $T$, you pick a node $u$, and you substitute it by a tree $T'$ by attaching the leaves of $T'$ to the children of $u$ in an injective manner. This means that the trees are closed under this operation of substitution, and thus we could expect that it preserves the property you're interested in. I admit though that I have no idea of what I'm doing with this defn, this might be a joke so please check on your side :)