Homeomorphic embedding of a rooted tree ($T⪯S$)

For finite rooted trees in the ancestor partial order, a rooted-tree homeomorphic embedding is an injective vertex map preserving lowest common ancestors: $h(u\wedge v)=h(u)\wedge h(v)$. The image of the source root need not be the host root. Edge paths and branching are preserved, while paths may be subdivided in the larger tree.

For rooted trees labelled in a preorder $Q$, a label-monotone tree embedding is a homeomorphic embedding of a rooted tree whose vertex map satisfies ${\ell }_T(v){\le }_Q{\ell }_S(h(v))$. The root need not map to the host root. For ordered child lists, require in addition that the map preserve the relative order of distinct branches.