A substring α of the tree’s upper frontier that matches some production A → α where reducing α to A is one step in the reverse of a rightmost derivation. We call such a string a handle.
a handle of a right-sentential form γ is a production A → β and a position in γ where β may be found and replaced by A to produce the previous right-sentential form in a rightmost derivation of γ i.e.,
if S ⇒∗ rm αAw ⇒rm αβw then A → β in the position following α is a handle of αβw
Because γ is a right-sentential form, the substring to the right of a handle contains only terminal symbols.
This all is the formal definition.
Informally you can understand handle like this:
for a given production, if a string is given to reduce, you start replacing the string with the right-hand side of production, the item that is removed or replaced is called handle. Reduce the given string until you get the start symbol.
1. S -> aABe
A-> Abc /b
The string is abbcde find the handles.
aAbcde // after replacing the second 'b' with 'A' from left So b is one of handle
aAde // after replacing 'Abc' with A So Abc is the second handle.
aABe // after replacing 'd' with 'B' So, 'd' is third handle.
S // after replacing 'aABe' with 'S', So 'aABe' is fourth handle.
2. E -> E+n / E * n / n
find the handles for n+n*n
E+n*n // replace first n with E so n is first handle.
E*n // replace E+n with E so E+n is the second handle.
E // replace E*n with E so E*n is third handle.
So handles are n, E+n , E*n