An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2ldots v_k$ 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 called decreasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,ldots,k-2$. We say 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 root of 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 are drawn as different points in the plane and its edges are straight line segments. Let $overline{alpha}(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing path of length $overline{alpha}(G)$. Let $overline{tau}(G)$ be the maximum integer such that every edge-ordered straight-line drawing of $G$ %under any edge labeling contains a monotone non-crossing complete binary tree of size $overline{tau}(G)$. In this paper we show that $overline alpha(K_n) = Omega(loglog n)$, $overline alpha(K_n) = O(log n)$, $overline tau(K_n) = Omega(loglog log n)$ and $overline tau(K_n) = O(sqrt{n log n})$.
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