In a $\text{B}^+$- tree where each node can hold at most four key values, a root to leaf path consists of the following nodes: $$A = (49, 77, 83, -), B = (7, 19, 33, 44), C = (20*, 22*, 25*, 26*)$$ The $*$-marked keys signify that these are data entries in a leaf. Assume that a pointer between keys $k_1$ and $k_2$ points to a subtree containing keys in $[k_1, k_2)$, and that when a leaf is created, the smallest key in it is copied up into its parent. A record with key value 23 is inserted into the $\text{B}^+$- tree. The smallest key value in the parent of the leaf that contains $25*$ is _________ . (Answer in integer)

Initial State:

• Node $A = (49, 77, 83, -)$ (Root)

• Node $B = (7, 19, 33, 44)$ is an internal node.

• Leaf node $C = (20^*, 22^*, 25^*, 26^*)$ lies in the subtree between keys $19$ and $33$, which holds keys in $[19, 33)$.

Insert key $23$ into leaf node $C$:

• Adding $23^*$ to node $C$ gives: $(20^*, 22^*, 23^*, 25^*, 26^*)$, which has 5 keys (exceeds maximum capacity of 4 keys).

• Leaf node $C$ splits into two leaf nodes:

  • Left leaf $C_1 = (20^*, 22^*, 23^*)$

  • Right leaf $C_2 = (25^*, 26^*)$

• The smallest key in the right leaf $C_2$, which is $25$, is copied up to the parent node $B$.

Insert copied-up key $25$ into internal node $B$:

• Node $B$ currently has keys $(7, 19, 33, 44)$.

• Inserting $25$ makes the keys in $B$: $(7, 19, 25, 33, 44)$, which has 5 keys and overflows.

• In an internal node split with 5 keys, the middle key ($25$) is pushed up to the parent (Root $A$).

• Node $B$ splits into:

  • Left internal node $B_1 = (7, 19)$

  • Right internal node $B_2 = (33, 44)$

**Identify the parent of the leaf containing $25^*$:**

• The leaf containing $25^*$ is $C_2 = (25^*, 26^*)$, covering range $[25, 33)$.

• This is the first child of the new right internal node $B_2 = (33, 44)$.

• The keys in $B_2$ are $33$ and $44$.

• The smallest key value in this parent node ($B_2$) is 33.