Several alternative restrictions on derivation trees could yield similar results for demonstrating metalinearness or regularity:
Bounded Branching: Restricting the number of branching nonterminal nodes in the derivation tree can help control the complexity of the generated language. For instance, a tree could be defined as "bounded-branching" if it allows a maximum of k branching nodes, which could facilitate the demonstration of k-linearity.
Context-Free Path Restrictions: Imposing conditions on the paths within the derivation tree, such as limiting the number of context-free rules that can be applied consecutively, may help in establishing regularity. For example, if a path can only contain a certain number of context-free applications before a non-context-free rule must be applied, this could lead to a clearer classification of the language.
Layered Derivation Trees: Introducing a layered structure to the derivation tree, where each layer corresponds to a specific type of rule (context-free, non-context-free), could help in analyzing the generative power of the grammar. This layered approach would allow for a more granular examination of how different types of rules interact and contribute to the overall language.
Nonterminal Degree Constraints: Restricting the degree of nonterminal nodes (i.e., the number of children a nonterminal can have) could also be beneficial. For instance, allowing only binary branching (each nonterminal can have at most two children) may simplify the analysis and help in proving regularity.
Contextual Independence: Defining stricter conditions for contextual independence among nonterminal nodes could lead to clearer boundaries between different language classes. For example, ensuring that no two nonterminal nodes in neighboring paths can share a common context could help in establishing regularity.
By exploring these alternative restrictions, researchers can potentially uncover new pathways for demonstrating metalinearness and regularity in various grammar formalisms.