Let T be a binary search tree containing integer numbers and let x be an integer not contained in T$.$ Consider the following two statements:

$$ Statement $1$: The successor of x has no left child.

$$ Statement $2$: The predecessor of x has no right child.

Which of the following logical implications are correct regarding those$$statements$?$

Select one or more:

$$ a$.$ If Statement $2$ is true, then Statement $1$ is true.

$$ b$.$ If Statement $2$ is true, then Statement $1$ is false.

$$ c$.$ If Statement $1$ is true, then Statement $2$ is true.

$$ d$.$ If Statement $1$ is true, then Statement $2$ is false.

$$ e$.$ If Statement $1$ is false, then Statement $2$ is false.

$$ f$.$ If Statement $2$ is false, then Statement$1$is$$false$.$