Use the fact that awk splits the lines in fields based on a field separator, that you can define. Hence, defining the field separator to / you can say:

awk -F "/" '{print $NF}' input

as NF refers to the number of fields of the current record, printing $NF means printing the last one.

So given a file like this:

/home/parent/child1/child2/child3/filename
/home/parent/child1/child2/filename
/home/parent/child1/filename

This would be the output:

$ awk -F"/" '{print $NF}' file
filename
filename
filename