Step 1: Examine each statement.
Statement I is correct.
Carrot belongs to the family Apiaceae, in which the inflorescence is
\[
\boxed{\text{Umbel}.}
\]
Statement II is correct.
The male flowers of Ficus possess
\[
\boxed{\text{three stamens}.}
\]
Statement III is incorrect.
Pneumatophores of mangrove plants such as Rhizophora are
\[
\boxed{\text{negatively geotropic}}
\]
as they grow upward for gaseous exchange.
Statement IV is incorrect.
In grapevine, the stem gives rise to tendrils opposite the leaves.
The axillary bud normally develops into a branch and not into a tendril.
Step 2: Choose the correct statements.
Hence,
\[
\boxed{\text{Statements I and II only are correct}.}
\]
Therefore, the correct option is
\[
\boxed{(A).}
\]