The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Bra\ss-K\'arolyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\cal T}$ and $ex_{\rightarrow}(n,T) = \Omega(n\log n)$ for $T \not\in {\cal T}$. We also describe the family of the convex geometric trees with linear Tur\' an number and show that for every convex geometric tree $F$ not in this family, $ex_{\cir}(n,F)= \Omega(n\log \log n)$.

中文翻译：

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具有线性极值函数的有序和凸几何树

许多研究人员广泛研究了用于有序和凸几何无环图 $F$ 的极值函数 $ex_{\rightarrow}(n,F)$ 和 $ex_{\cir}(n,F)$。基本问题是确定何时 $ex_{\rightarrow}(n,F)$ 和 $ex_{\cir}(n,F)$ 在 $n$ 中是线性的，后者由 Bra\ss-K\'arolyi 提出-Valtr 在 2003 年。在本文中，我们为每棵树 $F$ 回答这两个问题。我们给出了具有 $k$ 边的有序树族 $\cal T$ 的禁止子图表征，并证明 $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2 }$ 对于所有 $n \geq k + 1$ 当 $T \in {\cal T}$ 和 $ex_{\rightarrow}(n,T) = \Omega(n\log n)$ 对于 $T \not \in {\cal T}$。我们还用线性 Tur\' 数字描述了凸几何树的族，并表明对于每个不在这个族中的凸几何树 $F$，