A processor has $$16$$ integer registers $$\left( {R0,\,\,R1,\,\,..\,\,,\,\,R15} \right)$$) and $$64$$ floating point registers $$(F0, F1,… , F63).$$ It uses a $$2$$-byte instruction format. There are four categories of instructions: Type-$$1,$$ Type-$$2,$$ Type-3, and Type-$$4.$$ Type-$$1$$ category consists of four instructions, each with $$3$$ integer register operands $$(3Rs)$$. Type-$$2$$ category consists of eight instructions, each with $$2$$ floating point register operands $$(2Fs).$$ Type-$$3$$ category consists of fourteen instructions, each with one integer register operand and one floating point register operand $$(1R+1F).$$ Type-$$4$$ category consists of $$N$$ instructions, each with a floating point register operand $$(1F).$$

The maximum value of $$N$$ is __________.

A queue is implemented using a non-circular singly linked list. The queue has a head pointer and a tail pointer, as shown in the figure. Let $$n$$ denote the number of nodes in the queue. Let $$enqueue$$ be implemented by inserting a new node at the head, and $$dequeue$$ be implemented by deletion of a node from the tail.

Which one of the following is the time complexity of the most time-efficient implementation of $$enqueue$$ and $$dequeue,$$ respectively, for this data structure?

The postorder traversal of a binary tree is $$8,9,6,7,4,5,2,3,1.$$ The inorder traversal of the same tree is $$8,6,9,4,7,2,5,1,3.$$ The height of a tree is the length of the longest path from the root to any leaf. The height of the binary tree above is ______.

Let $$G$$ be a simple undirected graph. Let $${T_D}$$ be a depth first search tree of $$G.$$ Let $${T_B}$$ be a breadth first search tree of $$G.$$ Consider the following statements.

$$(I)$$ No edge of $$G$$ is a cross edge with respect to $${T_D}.$$ ($$A$$ cross edge in $$G$$ is between
$$\,\,\,\,\,\,\,\,$$ two nodes neither of which is an ancestor of the other in $${T_D}.$$)
$$(II)$$ For every edge $$(u,v)$$ of $$G,$$ if $$u$$ is at depth $$i$$ and $$v$$ is at depth $$j$$ in $${T_B}$$, then
$$\,\,\,\,\,\,\,\,\,\,\,$$ $$\left| {i - j} \right| = 1.$$

Which of the statements above must necessarily be true?