Derivation tree

Derivation tree --- дерево вывода.

Let $G$ be a CF-grammar and $x$ be a sentence form in $G$.

All equivalent derivations of $x$ can be represented by its derivation tree that is an ordered rooted tree labeled by elements from $V\bigcup \{e\}$, such that the following properties hold. The root of the tree has a label $S$, labels of all leaves enumerated according to their ordering forms $x$ and for any internal node $q$ of the tree with a list of all sons $q_1, q_2, \ldots, q_r,$ $r\geq 0,$ enumerated according to their ordering, there is such a production $A\rightarrow a_1 a_2 \ldots a_r$ of the grammar that either $a_1 a_2 \ldots a_r \not = e$ and the nodes $q, q_1, q_2,$ $\ldots, q_r$ have labels $A, a_1, a_2, \ldots, a_r$, respectively, or $a_1 a_2 \ldots a_r = e$, $r=1$, $q$ has a label $A$ and $q_1$ has the empty string $e$ as a label.

Other names are Parse tree, Syntax tree.