the cohomology groups of coherent sheaves on rational trees.

A [rational tree]{} $C$ is a connected, projective, nodal curve of arithmetic genus zero. If $C$ is a rational tree, we call a component $E$ of $C$ an [end]{} if $E$ contains at most one node of $C$.

Given a connected projective nodal curve $C$, define the dual graph of $C$ to be the graph $\Gamma _C$ whose vertices are indexed by the components $C_i$ of $C$ and whose edges between the distinct vertices $[C_i]$ and $[C_j]$ are indexed by $\{ p\in C_i \cap C_j \}$.

[*Remark*]{}. A connected projective nodal curve $C$ is a rational tree if and only if all its components are smooth rational curves and its dual graph $\Gamma _C$ is a tree (for a proof see [@De]).

\[basspl\] Let $C$ be a rational tree, and let $\nu : \tilde C \rightarrow C$ be the normalization of $C$ at the points $\{p_1, \ldots , p_r \} \subset Sing(C)$; denote by $\iota : \{p _1, 
\ldots , p_r \} \hookrightarrow C$ the inclusion morphism. For any locally free sheaf ${\mathcal F}$ of finite rank on $C$ we have the following short exact sequence of sheaves on $C$: $$0 