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An edge-ordered graph is a graph with a total ordering of its edges. A path$P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if$(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; it is calleddecreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$. Wesay that $P$ is monotone if it is increasing or decreasing. A rooted tree $T$in an edge-ordered graph is called monotone if either every path from the rootof to a leaf is increasing or every path from the root to a leaf is decreasing.  Let $G$ be a graph. In a straight-line drawing $D$ of $G$, its vertices aredrawn as different points in the plane and its edges are straight linesegments. Let $\overline{\alpha}(G)$ be the maximum integer such that everyedge-ordered straight-line drawing of $G$ %under any edge labeling contains amonotone non-crossing path of length $\overline{\alpha}(G)$. Let$\overline{\tau}(G)$ be the maximum integer such that every edge-orderedstraight-line drawing of $G$ %under any edge labeling contains a monotonenon-crossing complete binary tree of size $\overline{\tau}(G)$. In this paperwe show that $\overline \alpha(K_n) = \Omega(\log\log n)$, $\overline\alpha(K_n) = O(\log n)$, $\overline \tau(K_n) = \Omega(\log\log \log n)$ and$\overline \tau(K_n) = O(\sqrt{n \log n})$.

Non-crossing Monotone Paths and Binary Trees in Edge-ordered Complete Geometric Graphs