derivation tree

Given a formal grammar $G=(\mathrm{\Sigma },N,P,\sigma )$, recall that a derivation from words $u$ to $v$ over $\mathrm{\Sigma }$ can be visualized as a finite sequence of words over $\mathrm{\Sigma }$, connected by the binary relation $\Rightarrow$:

$$u_0\Rightarrow u_1\Rightarrow \mathrm{\cdots }\Rightarrow u_n$$ (1)

where $u=u_0$ and $v=u_n$. Each $u_i\Rightarrow u_{i+1}$ is a derivation step, which means that there is a production in $P$ which, when applied to $u_i$, yields $u_{i+1}$. In other words, there is $A\to B$ in $P$ such that $u_i=xAy$ and $u_{i+1}=xBy$, where $x,y$ are words over $\mathrm{\Sigma }$.

When the formal grammar $G$ is context-free, a derivation can be represented by an ordered tree, revealing the structure behind the derivation that is usually not apparent in the linear representation $(1)$ above. This ordered tree is variously known as a derivation tree or a parse tree, depending how it is being used.

In the foregoing discussion, $G$ is context-free, and any derivation of $G$ begins with $\sigma$, the starting non-terminal.

Definition. A parse tree of $G$ is an ordered tree $T$ such that

1. the nodes of $T$ are labeled by elements of $\mathrm{\Sigma }$, or the empty word $\lambda$,

2. if a node with label $A$ has children $N_1,\mathrm{\dots },N_k$ such that , and that each $N_i$ has label $A_i$, then $A\to A_1\mathrm{\cdots }{A}_k$ is a production of $P$.

A parse tree such that the root has label $\sigma$ is called a derivation tree, or a generation tree. Every subtree of a derivation tree is a parse tree.

Remark. Since $G$ is context-free, in a parse tree, any node that is not a leaf is labeled by a non-terminal symbol.

For example, if $\mathrm{\Sigma }=\{\sigma ,a,b,X,Y\}$, $N=\{\sigma ,X,Y\}$, and the productions of $P$ are

$$\sigma \to aXY,X\to bYb,Y\to Xa,Y\to a,$$

then

represents a derivation tree of $G$. The tree represents the following derivations

• $\sigma \Rightarrow aXY\Rightarrow abYbY\Rightarrow abXabY\Rightarrow abXabb$

• $\sigma \Rightarrow aXY\Rightarrow abYbY\Rightarrow abYbb\Rightarrow abXabb$

• $\sigma \Rightarrow aXY\Rightarrow aXb\Rightarrow abYbb\Rightarrow abXabb$

Definition. If ${\mathrm{\ell }}_1,\mathrm{\dots },{\mathrm{\ell }}_m$ are the leaves of a parse tree $T$, with , then the result of $T$ is the word $B_1\mathrm{\cdots }{B}_m$, where $B_i\in \mathrm{\Sigma }$ is the label of ${\mathrm{\ell }}_i$. A word over $\mathrm{\Sigma }$ is said to correspond to a parse tree if it is the result of the tree.

In the example above, the result of the tree is $abXabb$.

Remark. A derivable word may correspond to several derivation trees. See the entry ambiguous grammar for more detail.

Title	derivation tree
Canonical name	DerivationTree
Date of creation	2013-03-22 19:00:17
Last modified on	2013-03-22 19:00:17
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 68Q42
Classification	msc 68Q45
Synonym	generation tree
Defines	parse tree
Defines	result of a derivation tree