The number of ways in which the numbers $$1, 2, 3, 4, 5, 6, 7$$ can be inserted in an empty binary search tree, such that the resulting tree has height $$6,$$ is _____________.

$$Note:\,\,\,The\,\,height\,\,of\,\,a\,tree\,\,with\,\,a\,\,\sin gle\,\,node\,\,is\,\,0$$

$$N$$ items are stored in a sorted doubly linked list. For a $$delete$$ operation, a pointer is provided to the record to be deleted. For a $$decrease$$-$$key$$ operation, a pointer is provided to the record on which the operation is to be performed.

An algorithm performs the following operations on the list in this order:
$$\Theta \left( N \right),\,\,delete,\,\,O\left( {\log N} \right)\,insert,\,$$ $$\,O\left( {\log N} \right)\,fund, and $$ $$\Theta \left( N \right)\,$$ $$decrease$$-$$key.$$ What is the time complexity of all these operations put together?

Suppose a database schedule $$S$$ involves transactions $${T_1},\,...,\,{T_n}.$$ Construct the precedence graph of $$S$$ with vertices representing the transactions and edges representing the conflicts. If $$S$$ is serializable, which one of the following orderings of the vertices of the precedence graph is guaranteed to yield a serial schedule?

Consider the following database schedule with two transactions, T₁ and T₂

 S = r2(X); r1(X); r2(Y); w1(X); r1(Y); w2(X); a1; a2

where r_i(Z) denotes a read operation by transaction T_i on a variable Z, w_i(Z) denotes a write operation by T_i on a variable Z and a_i denotes an abort by transaction T_i .

Which one of the following statements about the above schedule is TRUE?