Parsing Tree and Derivation?

Question

I don't understand the relantioship between parsing tree and derivation. The parsing tree is invariant with respect to the derivation, but does this mean that regardless of the derivation (rightmost or leftmost) the parsing tree remains the same? Or according to the method (rightmost or leftmost) does the parsing tree change? Pls help me Sorry for my bad english.

Answer 1

The parse tree is a record of the derivation. Each non-leaf node of the tree is the result of a single derivation step.

At the root of the parse tree and at the beginning of the derivation, you find the grammar's start symbol. A derivation step replaces a non-terminal with the right-hand side of some production which has that non-terminal on the left-hand side. In the tree, the node corresponding to the non-terminal is given a sequence of children, each one a symbol in the right-hand side of the production. Terminal symbols become leaf nodes and non-terminals will eventually become the top of a subtree.

If the grammar is unambiguous, there is only one parse tree for each derivable sentence. But that parse tree represents a large number of possible derivations, unless the grammar is linear (that is, the right-hand side of every production contains at most one non-terminal). In a derivation which is being built, you can select any non-terminal for the next derivation step; in the parse tree, you can select any node representing a non-terminal which does not yet have children.

The leftmost and rightmost derivations are just two of these many possibilities. (Again, unless the grammar is linear, in which case the leftmost and rightmost derivations are the same derivation.) But a derivation doesn't have to be leftmost or rightmost. At each step, it can choose any non-terminal, not only the leftmost or rightmost one. In the tree representation, a possible derivation can be generated by any valid topological sort of the nodes.

But that doesn't matter in practical terms. The only useful practical question is whether there is more than one different parse tree for each sentence in the grammar, which is exactly the same as asking whether there is more than one leftmost derivation or more than one rightmost derivation.