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The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ forordered and convex geometric acyclic graphs $F$ have been extensivelyinvestigated 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 latterposed by Bra\ss-K\'arolyi-Valtr in 2003. In this paper, we answer both thesequestions for every tree $F$.  We give a forbidden subgraph characterization for a family $\cal T$ ofordered 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 alsodescribe the family of the convex geometric trees with linear Tur\' an numberand show that for every convex geometric tree $F$ not in this family,$ex_{\cir}(n,F)= \Omega(n\log \log n)$.

Ordered and convex geometric trees with linear extremal function