The rooted tree embedding problem into points in the plane (Q1314442)

Let \(T\) be a rooted tree with \(n\) vertices and \(N\) a set \(\{p_ 1,\dots,p_ n\}\) of \(n\) points in general position in \(\mathbb{R}^ 2\). Denote the sets of vertices and edges of \(T\) by \(V(T)\) and \(E(T)\), respectively. A bijection \(\varphi\) from \(V(T)\) to \(N\) satisfying the conditions: (C1) \(\varphi(r_ 1)=p_ 1\) \((r_ 1\) is the root of \(T)\), (C2) for nonadjacent edges \(u_ 1v_ 1\), \(u_ 2v_ 2 \in E(T)\), the line segments \(\overline {\varphi (u_ 1) \varphi(v_ 1)}\), \(\overline {\varphi (u_ 2) \varphi (v_ 2)}\) are disjoint, is called a rooted tree embedding (or \(rt\)-embedding). The main theorem states that an \(rt\)-embedding of \(V(T)\) on \(N\) always exists and that some \(rt\)-embedding can be constructed in polynomial time with respect to \(n\).