= Solution
Define the directed successor relation by $E(u,v)\iff l(u)=c(v)\lor r(u)=c(v)$. Then the required <predicate> is
$$
\boxed{\forall u,v\in T.\quad E(u,v)\Rightarrow v(u)<v(v),}
$$
where $v(u)$ denotes the natural-number information component of node $u$. In the source's indexed notation this is $\forall n_i,n_j\in T,\ (l_i=c_j\lor r_i=c_j)\Rightarrow v_i<v_j$.
This formulation checks both successor links against every node with the corresponding identifier, so it does not silently assume part (d)'s uniqueness condition. It checks existing successors; if every identifier must denote an actual node, add the separate <referential integrity> condition $\forall u\in T,\ \exists v,w\in T,\ c(v)=l(u)\land c(w)=r(u)$, or exempt an explicitly designated null identifier. The source specifies no null convention. Indeed a nonempty finite structure with two actual successors at every node cannot have strictly increasing values along all edges: following successors would eventually create a <directed cycle>. The implication above remains meaningful for leaves represented by absent or null successors.
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