Consider the following recurrence :

f(1) = 1;

f(2n) = 2f(n) $$-$$ 1, for n $$\ge$$ 1;

f(2n + 1) = 2f(n) + 1, for n $$\ge$$ 1;

Then, which of the following statements is/are TRUE?

Let G(V, E) be a directed graph, where V = {1, 2, 3, 4, 5} is the set of vertices and E is the set of directed edges, as defined by the following adjacency matrix A.

$$A[i][j] = \left\{ {\matrix{ {1,} & {1 \le j \le i \le 5} \cr {0,} & {otherwise} \cr } } \right.$$

A[i][j] = 1 indicates a directed edge from node i to node j. A directed spanning tree of G, rooted at r $$\in$$ V, is defined as a subgraph T of G such that the undirected version of T is a tree, and T contains a directed path from r to every other vertex in V. The number of such directed spanning trees rooted at vertex 5 is _____________.

Which one of the following statements is TRUE?

Consider the augmented grammar with {+, *, (, ), id} as the set of terminals.

S' $$\to$$ S

S $$\to$$ S + R | R

R $$\to$$ R * P | P

P $$\to$$ (S) | id

If I₀ is the set of two LR(0) items {[S' $$\to$$ S.], [S $$\to$$ S. + R]}, then goto(closure(I₀), +) contains exactly _________ items.