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1. Ali volunteered to pick the mangoes and threw them down to me under the tree.

or

2. Ali volunteered to pick the mangoes and throw them down to me under the tree.

I have problems identifying the verb tense in a complex sentence.

I tend to assume that "volunteered" is the main verb controlling "pick and throw" that follow thus rendering throw in base form.

Is it correct?

Eg: She would take her mother at the shopping mall and send her home after that.
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Your assumption is correct.

Ali volunteered to

pick the mangoes
and ( to )
throw them down to me under the tree.

CJ
Comments  
It really depends on the writer's intent. Did Ali volunteer to pick up the mangoes, afterwhich he threw them down to me under the tree? If that is the case, then the first statement is correct as it stands.

The second sentence could also be correct; however it has a different meaning altogether. Ali volunteered to pick the mangoes as well as volunteered to throw them down to me under the tree.